ON THE RELATIONSHIP BETWEEN

TENSOR PRODUCTS AND JOINT SPECTRA

Enrico Boasso

Contents

1. Introduction 2

2. Spectra of tensor products of operators 3

3. Banach spaces complexes 5

4. Joint spectra of commutative tuples of operators 6

5. Tensor products and joint spectra of commuta-

tive tuples of operators 8

6. Lie algebras 10

7. Joint spectra of representations of Lie algebras 11

8. An axiomatic tensor product 15

9. The main results 17

10. Applications 29

11 References 32

1

2

1. Introduction.

In this talk we deal with several joint spectra defined for representations

of complex solvable finite dimensional Lie algebras in complex Banach

spaces. The joint spectra to be considered are all related to the Taylor

joint spectrum. In fact, they are the Taylor, the S lodkowski, the Fredholm,

the split and the Fredholm split joint spectra. The main concern of this

talk is to present the behavior of these joint spectra with respect to the

procedure of passing from two given such representations,

ρ1: L1 → L(X1), ρ2: L2 → L(X2),

to the tensor product representation of the direct sum of the algebras,

ρ: L1 × L2 → L(X1⊗X2), ρ = ρ1 ⊗ I + I ⊗ ρ2,

where X1⊗X2 is a precise tensor product of the Banach spaces X1 and X2,

and I denotes the identity operator of both X1 and X2. In addition, it is

also described the spectral contributions of ρ1 and ρ2 to the aforementioned

joint spectra of the multiplication representation

ρ: L1 × Lop2 → L(J), ρ(T ) = ρ1(l1)T + Tρ2(l2),

where J ⊆ L(X2, X1) belongs to a class of operator ideals between the

Banach spaces X1 and X2, and Lop2 is the opposite algebra of L2. However,

in order to accurately pesent the main results of this talk, it is first reviewed

how the theory of tensor product is placed within the general theory of

joint spectra.

3

2. Spectra of tensor products of operators.

The characterization of joint spectra of operators or families of operators

constructed from tensor products of linear and bounded Banach space

maps has a rich history and a wide development, which can be traced

back to a work by A. Brown and C. Pearcy (Proc. Amer. Math. Soc.

1966). In this work, given a Hilbert space H and A and B ∈ L(H), it was

proved that

σ(A⊗B) = σ(A) · σ(B) = {a · b : a ∈ σ(A), b ∈ σ(b)},

where A⊗B ∈ L(H1⊗H2), and where H1⊗H2 denotes the completion of

the algebraic tensor product H1 ⊗H2 with respect to its canonical scalar

product.

It is important to recall that the arguments that proves the above for-

mula shows the intimate connection between the operator A⊗B and the

multiplication operator

X → AXB

on the Hilbert-Schmidt class. In fact, the spectra of A⊗B was presented as

the spectra of a multiplication operator of the above type. Consequently,

from the beginning of the theory, not only spectra of tensor products

of operators were characterized, but also spectra of multiplicaiton maps

between operator ideals were studied.

As regard spectra of tensor products of Banach space operators, the

result of A. Brown and C. Pearcy was generalized to Banach space, arbi-

trary polynomials and some cross norms. In fact, M. Schechter (J. Funct.

Anal. 1969) proved that

σ(P (A⊗ I, I ⊗B)) = P (σ(A), σ(B)) = {P (a, b) : a ∈ σ(A), b ∈ σ(B)},

where P is a polynomial in two variables and I the identity map of the

corresponding Banach spaces. Actually, in the work by Schechter it was

proved a more general result but this formula is enough for our purposes.

In many other works several joint spectra of bounded and linear maps

constructed from tensor products of Banach space operators defined in the

completion of the algebraic tensor product of Banach spaces with respect

to some cross were studied. To mention only some of the articles and

4

author in the field, recall the papers by M. Reed and B. Simon (1973), M.

Schechter and M. Snow (1975), and T. Ichinose (1972, 1978 I and II). It is

woth to recall that in all these works the operators A⊗ I and I ⊗B were

central.

In several variable operator theory only joint spectra related to the

Taylor joint spectra will be considered. However, in order to introduce

the definition of these joint spectra, first it will be reviewed the notion of

complex of Banach spaces.

5

3. Banach space complexes.

Let X and Y be two Banach spaces. In this talk L(X, Y ) denotes the

algebra of all bounded and linear maps from X to Y , and K(X, Y ) the

closed ideal of all compact operators of L(X, Y ). As usual, if X = Y ,

then L(X) = L(X, X) and K(X) = K(X, X). If T ∈ L(X, Y ), then R(T )

and N(T ) denote the range and the null space of T respectively, that is

R(T ) = {y ∈ Y : there is x ∈ X such that y = T (x)} and N(T ) = {x ∈X : T (x) = 0}.

A finite complex of Banach space and bounded linear operators (X, d)

is a sequence

0→ Xndn−→ Xn−1 → . . .→ X1

d1−→ X0 → 0,

where n ∈ N, Xp is a Banach space, and the maps dp ∈ L(Xp, Xp−1) are

such that dp ◦ dp−1 = 0, for p = 0, . . . , n.

For a fixed integer p, 0 ≤ p ≤ n, (X, d) is said Fredholm split in degree

p if there are continous linear operators

Xp+1hp←− Xp

hp−1←−−− Xp−1,

such that dp+1hp + hp−1dp = Ip − kp, where Ip denotes the identity map

of Xp and kp is a compact operator defined in Xp. In addition, if kp ≡ 0,

then the complex (X, d) is called split in degree p.

If (X, d) is a finite complex of Banach spaces, the p-homology group of

(X, d) is the set Hp(X, d) = N(dp)/R(dp+1), p = 0, . . . , n.

Next consider tensor products of complex.

Let (X, d′) and (Y, d′′) be complexes of Banach spaces

0→ Xnd′n−→ Xn−1 → . . .→ X1

d′1−→ X0 → 0,

0→ Ymd′′m−−→ Ym−1 → . . .→ Y1

d′′1−→ Y0 → 0.

Suppose that are defined both the tensor products Xp⊗Yq and the di-

agrams

Xp−1⊗Yq

d′p⊗Iq←−−−− Xp⊗Yq

(−1)p−1Ip−1⊗d”q

y y(−1)pIp⊗d”q

Xp−1⊗Yq−1 ←−−−−−−d′

p⊗Iq−1

Xp⊗Yq−1,

6

where Ip (resp. Iq) denote the identity map of Xp (resp. Yq), and p =

0, . . . , n, q = 0, . . . m. Then it is possible to consider the complex of

Banach space (X⊗Y, d), where

X⊗Yk = ⊕p+q=kXp⊕Yq,

and dk: X⊗Yk → X⊗Yk−1 is the map

dk: Xp⊗Yq → Xp−1⊗Yq ⊕Xp⊗Yq−1, dk = d′p ⊗ Iq + ηp ⊗ d′′,

where ηp: Xp → Xp is the map η = (−1)pIp, and k = p + q.

7

4. Joint spectra of commutative tuples of operators.

Let X be a complex Banach space and a = (a1, . . . , an) an n-tuple of

commuting operators. The Koszul complex of the tuple a is the sequence

of spaces and maps (X⊗∧(Cn), d), where ∧(Cn) is the exterior algebra of

Cn and d is the map defined by

dp(x⊗ 〈x1 ∧ · · · ∧ xp〉) =p∑

k=1

(−1)k+1a(x)⊗ 〈x1 ∧ . . . ∧ xk ∧ . . . ∧ xp〉,

whereˆmeans deletion. For p such that p ≤ 0 or p ≥ n + 1, dp ≡ 0.

Let λ = (λ1, . . . , λn) be an n-tuple of complex numbers. Then a− λ =

(a1 − λ1I, . . . , an − λnI) is a commutative tuple of operators, where I

denotes the identity of X. Consider the Koszul complex of the tuple a−λ,

that is the complex (X ⊗ ∧(Cn), d(λ)):

0→ X ⊗ ∧n(Cn)dn(λ)−−−→ X ⊗ ∧n−1(Cn)→ . . .→ X ⊗ ∧kCn dk(λ)−−−→ X⊗

∧k−1 (Cn)→ . . . . . .→ X ⊗ ∧1(Cn)d1(λ)−−−→ X ⊗ ∧0(Cn)→ 0.

Define the sets

σp(a) = {λ ∈ Cn : Hp(X ⊗ ∧Cn, d(λ)) 6= 0},

σp,e(a) = {λ ∈ Cn : Hp(X ⊗ ∧Cn, d(λ)) =∞}.

spp(a) = {λ ∈ Cn : Hp(X ⊗ ∧Cn, d(λ)) is not split in degree p},

spp,e(a) = {λ ∈ Cn : Hp(X ⊗ ∧Cn, d(λ)) is not Fredholm split in degree p}.

The Taylor joint spectrum of a is the set

σ(a) =n⋃

p=0

σp(a) = {λ ∈ Cn : H∗(X ⊗ ∧Cn, d(λ)) 6= 0}.

Moreover, the k-th δ-S lodkowski joint spectrum of a is the set

σδ,k(a) =k⋃

p=0

σp(a),

and the k-th π-S lodkowski joint spectrum of a is the set

σπ,k(a) =n⋃

p=n−k

σp(a) ∪ {λ ∈ Cn : R(dn−k(λ)) is not closed},

8

for 0 ≤ k ≤ n.

The Fredholm, or essential Taylor, joint spectrum of a is the set

σe(a) =n⋃

p=0

σp,e(a).

Furthermore, the k-th Fredholm or essential δ-S lodkowski joint spectrum

of a is the set

σδ,k,e(a) =k⋃

p=0

σp,e(a),

and the k-th Fredholm or essential π-S lodkowski joint spectrum of a is the

set

σπ,k,e(a) =n⋃

p=n−k

σp,e(a) ∪ {λ ∈ Cn : R(dn−k(a)) is not closed},

for 0 ≤ k ≤ n.

The split joint spectrum of a is the set

sp(a) =n⋃

p=0

spp(a).

In addition, the k-th δ-split joint spectrum of a is the set

spδ,k(a) =k⋃

p=0

spp(a),

and the k-th π-split joint spectrum of a is the set

spπ,k(a) =n⋃

p=n−k

spp(a),

for 0 ≤ k ≤ n.

The Fredholm or essential split joint spectrum of a is the set

spe(a) =n⋃

p=0

spp,e(a).

Moreover, the k-th Fredholm or essential δ-split joint spectrum of a is the

set

spδ,k,e(a) =k⋃

p=0

spp,e(a),

and the k-th Fredholm or essential π-split joint spectrum of a is the set

spπ,k,e(a) =n⋃

p=n−k

spp,e(a),

for 0 ≤ k ≤ n.

9

5. Tensor products and joint spectra of commutative tuples of

operators.

One of the most deeply studied problems within the theory of joint

spectra has been the determination of the spectral contributions that two

commuting systems of operator S = (S1, . . . , Sn) and T = (T1, . . . , Tm)

defined in the Banach spaces X1 and X2 respectively, make to the joint

spectra of the system (S⊗I, I⊗T ) = (S1⊗I, . . . , Sn⊗I, I⊗T1, . . . , I⊗Tm)

defined in X1⊗αX2, i.e., the completation of the algebraic tensor product

X1 ⊗ X2 with respect to a quasi-uniform crossnorm α, and where the

symbol I stands for the identity map both in X1 and X2. For example,

if X1 and X2 are Hilbert spaces and X1⊗X2 is the canonical completion

of X1⊗X2, then in a work by Z. Ceausescu and F. - H. Vasilescu (Proc.

Amer. Math. Soc. 1978) the Taylor joint spectrum of (S ⊗ I, I ⊗ T ) in

X1⊗X2 was characterized. Indeed, it was proved that

σ(S ⊗ I, I ⊗ T ) = σ(S)× σ(T );

see also the related work by the same authors (Studia Math. 1978).

In addition, since if S and T are Fredholm, then so is (S⊗ I, I ⊗T ) (C.

Grosu and F. - H. Vasilescu Integral Equations Operator Theory 1982),

σe(S ⊗ I, I ⊗ T ) = σe(S)× σ(T ) ∪ σ(S)× σe(T ).

On the other hand, in an operator ideal J ⊆ L(X2, X1) between the

Banach spaces X1 and X2, it is possible to consider tuples of left and right

multiplication: LS = (LS1 , . . . , LSn) and RT = (RT1 , . . . , RTm) respec-

tively, induced by commuting systems of operators S = (S1, . . . , Sn) and

T = (T1, . . . , Tm) defined in X1 and X2 respectively, where LU (A) = UA

and RV (B) = BV , U ∈ L(X1), V ∈ L(X2) and A, B ∈ J . However, as

in the case of a single operator, the tuple (LS , RT ) is closely related to

the system (S⊗ I, I⊗T′). Indeed, the canonical completion H⊗H

′of the

algebraic tensor product of a Hilbert space H and its dual can be regarded

as an operator ideal in L(H), (A. Brown and C. Pearcy Proc. Amer. Math

Soc. 1966). As regards this identification the operators Si ⊗ I and I ⊗ T′

j

correspond to the operators LSi and RTj respectively, for i = 1, . . . , n

and j = 1, . . . , m. In particular, the joint spectra of (LS , RT ) are closely

related to the corresponding joint spectra of (S ⊗ I, I ⊗ T ). In fact, in

10

the work by R. Curto and L. Fialkow (J. Funct. Anal. 71 (1987)) it was

proved that

σ(LS , LT ) = σ(S)× σ(T )

σe(LS , LT ) = σe(S)× σ(T ) ∪ σ(S)× σe(T ).

In the frame of Banach spaces, in a work by J. Eschmeier (J. reine

angew. Math. 1988) it was introduced an axiomatic tensor product which

is general and rich enough to allow, on the one hand, the description of the

Taylor, the split, the essential Taylor and the essential split joint spectra of

a system (S⊗I, I⊗T ) defined in the tensor product of two Banach spaces

and, on the other hand, the description of all the above-mentioned joint

spectra of tuples of left and right multiplications (LS , RT ) defined in a

class of operator ideals between Banach spaces introduced in the aforesaid

article.

In fact, it was proved that

σ(S)× σ(T ) ⊆ σ(S ⊗ I, I ⊗ T ) ⊆ sp(S ⊗ I, I ⊗ T ) ⊆ sp(S)× sp(T ),

σe(S)× σ(T ) ∪ σ(S)× σe(T ) ⊆ σe(S ⊗ I, I ⊗ T ) ⊆ spe(S ⊗ I, I ⊗ T )

⊆ spe(S)× sp(T ) ∪ sp(S)× spe(T ),

and

σ(S)× σ(T ) ⊆ σ(LS , LT ) ⊆ sp(LS , LT ) ⊆ sp(S)× sp(T ),

σe(S)× σ(T ) ∪ σ(S)× σe(T ) ⊆ σe(LS , LT ) ⊆ spe(LS , LT )

⊆ spe(S)× sp(T ) ∪ sp(S)× spe(T ).

However, all the other joint spectra related to the Taylor spectrum

have not described even in the Hilbert space. The main concern of this

talk consists in the study of these joint spectra in Banach spaces and

for representation of Lie algebras. The commutative case will be also

considered.

11

6. Lie Algebras.

A complex Lie algebra is a vector space over the complex field C provided

with a bilinear bracket, named the Lie product, [., .]: L × L → L, which

complies with the Lie conditions

[x, x] = 0, [[x, y], z] + [[y, z], x] + [[z, x], y] = 0,

for every x, y and z ∈ L. The second of these equations is called the Jacobi

identity.

An example of a Lie algebra structure is given by the algebra of all

bounded linear maps defined in a Banach space X, L(X), and the bracket

[., .]: L(X)× L(X)→ L(X), [S, T ] = ST − TS, for S and T ∈ L(X).

Given two Lie algebras L1 and L2 with Lie brackets [., .]1 and [., .]2respectively, a morphism of Lie algebras H: L1 → L2 is a linear map such

that H([x, y]1) = [H(x),H(y)]2, for x and y ∈ L1. In particular, when

L = L1 and L2 = L(X), X a Banach space, ρ: L → L(X) is said a

representation of the Lie algebra L.

Let L be a complex Lie algebra. A subspace I ⊆ L is called a subalgebra

of L if [I, I] ⊆ I, and an ideal if [I, L] ⊆ I, where if M and N are two

subsets of L, then [M,N ] denotes the set {[m,n] : m ∈ M, n ∈ N}. In

particular, L2 = [L,L] = {[x, y] : x, y ∈ L} is an ideal of L.

When L is a Lie algebra, a character of L is a linear map f : L→ C such

that f(L2) = 0, that is f : L→ C is a Lie morphism.

For any Lie algebra L there are two series of ideals. The derived series,

that is

L = L(1) ⊇ L(2) = [L,L] ⊇ L(3) = [L(2), L(2)] ⊇ . . . ⊇ L(k) = [L(k−1), L(k−1)],

and the descending central series, that is

L = L1 ⊇ L2 = L(2) = [L,L] ⊇ L3 = [L, L2] ⊇ . . . ⊇ Lk = [L,Lk−1] ⊇ . . . .

A Lie algebra L is named solvable (resp. nilpotent ) if there is some

positive integer k such that L(k) = 0 (resp. Lk = 0). Obviously a nilpotent

Lie algebra is solvable.

In this talk only complex solvable finite-dimensional Lie algebras are

considered.

12

7. Joint spectra of representations of Lie algebras.

Let L denote a complex solvable finite-dimensional Lie algebra, X a

complex Banach space and ρ: L→ L(X) a representation of L in X. The

Koszul complex of the representation ρ, is the sequence of Banach spaces

and bounded linear operators (X ⊗ ∧L, d(ρ)), where ∧L denotes the ex-

terior algebra of L, and dp(ρ): X ⊗∧pL→ X ⊗∧p−1L is the map defined

by

dp(ρ)(x⊗ 〈l1 ∧ · · · ∧ lp〉) =p∑

k=1

(−1)k+1ρ(lk)(x)⊗ 〈l1 ∧ . . . ∧ lk ∧ . . . ∧ lp〉

+∑

1≤i<j≤p

(−1)i+jx⊗ 〈[li, lj ] ∧ l1 ∧ . . . ∧ li ∧ . . . ∧ lj ∧ . . . ∧ lp〉,

whereˆmeans deletion. For p such that p ≤ 0 or p ≥ n + 1, n = dim L,

dp(ρ) ≡ 0.

If f is a character of L, then consider the representarion of L in X ρ−f ≡ρ−f ·I, where I denotes the identity map of X. Now, if H∗(X⊗∧L, d(ρ−f)) denotes the homology of the Koszul complex of the representation ρ−f ,

then define the sets

σp(ρ) = {f ∈ L∗ : f(L2) = 0,Hp(X ⊗ ∧L, d(ρ− f)) 6= 0},

σp,e(ρ) = {f ∈ L∗ : f(L2) = 0, dim Hp(X ⊗ ∧L, d(ρ− f)) =∞}.

spp(ρ) = {f ∈ L∗ : f(L2) = 0, (X ⊗ ∧L, d(ρ− f)) is not split in degree p},

spp,e(ρ) = {f ∈ L∗ : f(L2) = 0, (X ⊗ ∧L, d(ρ− f)) is not Fredholm split

in degree p}.

Consider the Koszul complex of the representation ρ− f , f a character

of L, that is

0→ X ⊗ ∧nLdn(ρ−f)−−−−−→ X ⊗ ∧n−1L→ . . .→ X ⊗ ∧kL

dk(ρ−f)−−−−−→ X ⊗ ∧k−1 → . . .

. . .→ X ⊗ ∧1Ld1(ρ−f)−−−−−→ X ⊗ ∧0L→ 0.

The Taylor joint spectrum of ρ is the set

σ(ρ) =n⋃

p=0

σp(ρ) = {f ∈ L∗ : f(L2) = 0,H∗(X ⊗ ∧L, d(ρ− f)) 6= 0}.

Moreover, the k-th δ-S lodkowski joint spectrum of ρ is the set

σδ,k(ρ) =k⋃

p=0

σp(ρ),

13

and the k-th π-S lodkowski joint spectrum of ρ is the set

σπ,k(ρ) =n⋃

p=n−k

σp(ρ)∪{f ∈ L∗ : f(L2) = 0, R(dn−k(ρ−f)) is not closed},

for 0 ≤ k ≤ n = dim L.

The Fredholm, or essential Taylor, joint spectrum of ρ is the set

σe(ρ) =n⋃

p=0

σp,e(ρ).

Furthermore, the k-th Fredholm or essential δ-S lodkowski joint spectrum

of ρ is the set

σδ,k,e(ρ) =k⋃

p=0

σp,e(ρ),

and the k-th Fredholm or essential π-S lodkowski joint spectrum of ρ is the

set

σπ,k,e(ρ) =n⋃

p=n−k

σp,e(ρ)∪{f ∈ L∗ : f(L2) = 0, R(dn−k(ρ)) is not closed},

for 0 ≤ k ≤ n = dim L.

The split joint spectrum of ρ is the set

sp(ρ) =n⋃

p=0

spp(ρ).

In addition, the k-th δ-split joint spectrum of ρ is the set

spδ,k(ρ) =k⋃

p=0

spp(ρ),

and the k-th π-split joint spectrum of ρ is the set

spπ,k(ρ) =n⋃

p=n−k

spp(ρ),

for 0 ≤ k ≤ n = dim L.

The Fredholm or essential split joint spectrum of ρ is the set

spe(ρ) =n⋃

p=0

spp,e(ρ).

14

Moreover, the k-th Fredholm or essential δ-split joint spectrum of ρ is the

set

spδ,k,e(ρ) =k⋃

p=0

spp,e(ρ),

and the k-th Fredholm or essential π-split joint spectrum of ρ is the set

spπ,k,e(ρ) =n⋃

p=n−k

spp,e(ρ),

for 0 ≤ k ≤ n.

Observe that

σδ,n(ρ) = σπ,n(ρ) = σ(ρ),

σe(ρ) = σδ,n,e(ρ) = σπ,n,e(ρ),

spδ,n(ρ) = spπ,n(ρ) = sp(ρ),

spδ,n,e(ρ) = spπ,n,e(ρ) = spe(ρ).

Next follows the properties of the above joint spectra.

Note that

σδ,k(ρ) ⊆ spδ,k(ρ), σπ,k(ρ) ⊆ spπ,k(ρ), σ(ρ) ⊆ sp(ρ),

σδ,k,e(ρ) ⊆ spδ,k,e(ρ), σπ,k,e(ρ) ⊆ spπ,k,e(ρ), σ(ρ, e) ⊆ sp(ρ, e).

Furthermore, when X is a Hilbert space, the above inclusions are equal-

ities.

Let X be a complex Banach space and σ a function which assigns a

compact non-void subset of the characters of L to each representation

ρ: L→ L(X) of a complex solvable finite dimensional Lie algebra L in X .

In addition, when L is a solvable (resp. nilpotent) Lie algebra, let I be an

ideal (resp. a subalgebra) of L. Now consider the representation ρ | I: I →L(X), that is the restriction of ρ to I. Then, σ has the projection property

, when for each ideal (resp. subalgebra) of a solvable (resp. nilpotent) Lie

algebra,

π(σ(ρ)) = σ(ρ | I),

where π: L∗ → I∗ is the restriction map.

Theorem: Let X be a complex Banach space, L a complex solvable finite

dimensional Lie algebra, and ρ: L → L(X) a representation of L in X. If

15

σ∗(ρ) denotes one of the aforementioned joint spectra of the representation

ρ, then σ∗(ρ) is a compact non-void subsets of L∗ that has the projection

property.

Let X be a Banach space and a = (a1, . . . , an) an n-tuple of commuting

operators defined in X. Let L be the commutative Lie subalgebra of L(X)

generated by the tuple a, that is L =< ai >1≤i≤n, and consider the

inclusion as a representation of L in L(X), that is ι: L→ L(X), ι(T ) = T ,

T ∈ L. Then, if as above σ∗ denotes one of the aforementioned joint

spectra, σ∗(ι) is reduced to the usual joint spectra for commutative tuples

of operators in the following way. If l = (l1, . . . , lm) is a basis of L, then

{(f(l1), . . . , f(lm)) : f ∈ σ} = σ(l1, . . . , lm), that is, in terms of the basis

l = (l1, . . . , lm) the joint spectrum σ coincides with the spectrum of the

n-tuple l.

A. S. Fainshtein and A. A. Dosiev studied the Functional Calculus and

the spectral mapping theorem for the Taylor and S lodkoswski joint spectra

of representations of Lie algebras.

16

8. An axiomatic tensor product.

The tensor product introduced by J. Eschmeier is recalled.

A pair 〈X, X〉 of Banach spaces will be called a dual pairing, if

(A) X = X ′ or (B) X = X ′.

In both cases, the canonical bilinear mapping is denoted by

X × X → C, (x, u) 7→ 〈x, u〉.

If 〈X, X〉 is a dual pairing, we consider the subalgebra L(X) of L(X)

consisting of all operators T ∈ L(X) for which there is an operator T ′ ∈L(X) with

〈Tx, u〉 = 〈x, T ′u〉,

for all x ∈ X and u ∈ X. It is clear that if the dual pairing is 〈X, X ′〉,then L(X) = L(X), and that if the dual pairing is 〈X ′, X〉, then L(X) =

{T ∗, T ∈ L(X)}. In particular, each operator of the form

fy,v: X → X, x 7→ 〈x, v〉y,

is contained in L(X), for y ∈ X and v ∈ X.

Definition. Given two dual pairings 〈X, X〉 and 〈Y, Y 〉, a tensor product

of the Banach spaces X and Y relative to the dual pairings 〈X, X〉 and

〈Y, Y 〉 is a Banach space Z together with continuous bilinear mappings

X × Y → Z, (x, y) 7→ x⊗ y; L(X)× L(Y ) 7→ L(Z), (T, S) 7→ T ⊗ S,

which satisfy the following conditions,

(T1) ‖ x⊗ y ‖=‖ x ‖‖ y ‖,

(T2) T ⊗ S(x⊗ y) = (Tx)⊗ (Sy),

(T3) (T1 ⊗ S1) ◦ (T2 ⊗ S2) = (T1T2)⊗ (S1S2), I ⊗ I = I,

(T4) Im(fx,u ⊗ I) ⊆ {x⊗ y: y ∈ Y }, Im(fy,v ⊗ I) ⊆ {x⊗ y: x ∈ X}.

Instead of Z it will be written X⊗Y instead of Z. Two applications

of this definition will be consider, namely, the completion X⊗αY of the

algebraic tensor product of the Banach spaces X and Y with respect to a

quasi-uniform crossnorm α, and an operator ideal between Banach spaces.

17

Let X and Y be Banach spaces. A crossnorm α is said to be quasi-

uniform with constant k on X ⊗ Y if

α((T ⊗ S)u) ≡‖ (T ⊗ S)u ‖α≤ k ‖ T ‖ ‖ S ‖ ‖ u ‖α,

for every pair (T, S) ∈ L(X)× L(Y ) and all u ∈ X ⊗ Y .

Next the tensor product representation is introduced.

Let L1 and L2 be two complex solvable finite dimensional Lie algebras,

X1 and X2 two complex Banach spaces, and ρi: Li → L(Xi), i = 1, 2, two

representations of Lie algebras. Suppose that there is a tensor product of

X1 and X2 relative to 〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2. Then it is possible

to consider the direct sum of the Lie algebras L1 and L2, L = L1 × L2,

which is a complex solvable finite dimensional Lie algebra, and the tensor

product representation of L in X1⊗X2, that is

ρ = ρ1 × ρ2: L→ L(X1⊗X2), ρ1 × ρ2(l1, l2) = ρ1(l1)⊗ I + I ⊗ ρ2(l2),

where I denotes the identity of X2 and X1 respectively.

The main concern of this talk is to study the joint spectra of the repre-

sentation ρ.

Another application of this axiomatic tensor product is multiplication

representation in an operator ideal between Banach spaces.

Definition. An operator ideal J between Banach spaces X2 and X1 is a

linear subspace of L(X2, X1) equipped with a space norm α such that

i) x1 ⊗ x2′ ∈ J and α(x1 ⊗ x2

′) =‖ x1 ‖‖ x2′ ‖,

ii) SAT ∈ J and α(SAT ) ≤‖ S ‖ α(A) ‖ T ‖,

for x1 ∈ X1, x2′ ∈ X2

′, A ∈ J , S ∈ L(X1), T ∈ L(X2), and x1 ⊗ x2′ is

the usual rank one operator X2 → X1, x2 7→< x2, x2′ > x1.

Recall that such an operator ideal J is naturally a tensor product rela-

tive to 〈X1, X1′〉 and 〈X2

′, X2〉, with the bilinear mappings

X1 ×X2′ → J, (x1, x2

′) 7→ x1 ⊗ x2′,

L(X1)× L(X2′)→ L(J), (S, T ′) 7→ S ⊗ T ′,

where S ⊗ T ′(A) = SAT .

18

Now, let L1 and L2 be two complex solvable finite dimensional Lie

algebras, X1 and X2 two complex Banach spaces, and ρi: Li → L(Xi),

i = 1, 2, two representations of Lie algebras. Consider the Lie algebra Lop2

and the adjoint representation ρ∗2: Lop2 → L(X2

′). Now, if L is the direct

sum of L1 and Lop2 , L = L1 × Lop

2 , then the multiplication representation

of L in J is

ρ: L→ L(J), ρ(l1, l2)(T ) = ρ1(l1)T + Tρ2(l2).

19

9. The main results.

Next the main theorems are stated.

Theorem A. Let L1 and L2 be two complex solvable finite dimensional

Lie algebras, X1 and X2 two complex Banach spaces, and ρi: Li → L(Xi),

i = 1, 2, two representations of Lie algebras. Suppose that there is a tensor

product of X1 and X2 relative to 〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2. Then,

if the tensor product representation of L = L1 × L2, ρ = ρ1 × ρ2: L →L(X1⊗X2) is considered,

i)⋃

p+q=k

σδ,p(ρ1)×σδ,q(ρ2) ⊆ σδ,k(ρ) ⊆ spδ,k(ρ) ⊆⋃

p+q=k

spδ,p(ρ1)×spδ,q(ρ2),

ii)⋃

p+q=k

σπ,p(ρ1)×σπ,q(ρ2) ⊆ σπ,k(ρ) ⊆ spπ,k(ρ) ⊆⋃

p+q=k

spπ,p(ρ1)×spπ,q(ρ2).

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

Theorem A’. Let X1 and X2 be two complex Banach spaces. Suppose

that there is a tensor product of X1 and X2 with respect to 〈X1, X1′〉 and

〈X2, X2′〉, X1⊗X2. Let a = (a1, . . . , an) and b = (b1, . . . , bm) be two

tuples of operators, ai ∈ L(X1), 1 ≤ i ≤ n, and bj ∈ L(X2), 1 ≤ j ≤ m,

such that the vector subspaces generated by them, 〈ai〉1≤i≤n and 〈bj〉1≤j≤m,

are nilpotent Lie subalgebras of L(X1) and L(X2) respectively. Consider

the (n+m)-tuple of operators defined in X1⊗X2, c = (a1⊗I, . . . , an⊗I, I⊗b1, . . . , I ⊗ bm), where I denotes the identity of X2 and X1 respectively.

Then

i)⋃

p+q=k

σδ,p(a)× σδ,q(b) ⊆ σδ,k(c) ⊆ spδ,k(c) ⊆⋃

p+q=k

spδ,p(a)× spδ,q(b),

ii)⋃

p+q=k

σπ,p(a)×σπ,q(b) ⊆ σπ,k(c) ⊆ spπ,k(c) ⊆⋃

p+q=k

spπ,p(a)×spπ,q(b).

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

20

Theorem B. Let X1 and X2 be two complex Banach spaces, L1 and L2

two complex solvable finite dimensional Lie algebras, and ρi: Li → L(Xi),

i = 1, 2, two representations of Lie algebras. Suppose that there is a

tensor product of X1 and X2 relative to 〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2.

Then, if the tensor product representation of the direct sum of L1 and L2,

ρ = ρ1 × ρ2: L1 × L2 → L(X1⊗X2) is considered,

i)⋃

p+q=k

σδ,p,e(ρ1)× σδ,q(ρ2)⋃ ⋃

p+q=k

σδ,p(ρ1)× σδ,q,e(ρ2) ⊆ σδ,k,e(ρ) ⊆

spδ,k,e(ρ) ⊆⋃

p+q=k

spδ,p,e(ρ1)× spδ,q(ρ2)⋃ ⋃

p+q=k

spδ,p(ρ1)× spδ,q,e(ρ2),

ii)⋃

p+q=k

σπ,p,e(ρ1)× σπ,q(ρ2)⋃ ⋃

p+q=k

σπ,p(ρ1)× σπ,q,e(ρ2) ⊆ σπ,k,e(ρ) ⊆

spπ,k,e(ρ) ⊆⋃

p+q=k

spπ,p,e(ρ1)× spπ,q(ρ2)⋃ ⋃

p+q=k

spπ,p(ρ1)× spπ,q,e(ρ2).

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

Theorem B’. Let X1 and X2 be two complex Banach spaces. Suppose

that there is a tensor product of X1 and X2 with respect to 〈X1, X1′〉 and

〈X2, X2′〉, X1⊗X2. Let a = (a1, . . . , an) and b = (b1, . . . , bm) be two

tuples of operators, ai ∈ L(X1), 1 ≤ i ≤ n, and bj ∈ L(X2), 1 ≤ j ≤ m,

such that the vector subspaces generated by them, 〈ai〉1≤i≤n and 〈bj〉1≤j≤m,

are nilpotent Lie subalgebras of L(X1) and L(X2) respectively. Consider

the (n+m)-tuple of operators defined in X1⊗X2, c = (a1⊗I, . . . , an⊗I, I⊗b1, . . . , I ⊗ bm), where I denotes the identity of X2 and X1 respectively.

Then

i)⋃

p+q=k

σδ,p,e(a)× σδ,q(b)⋃ ⋃

p+q=k

σδ,p(a)× σδ,q,e(b) ⊆ σδ,k,e(c) ⊆

spδ,k,e(c) ⊆⋃

p+q=k

spδ,p,e(a)× spδ,q(b)⋃ ⋃

p+q=k

spδ,p(a)× spδ,q,e(b),

ii)⋃

p+q=k

σπ,p,e(a)× σπ,q(b)⋃ ⋃

p+q=k

σπ,p(a)× σπ,q,e(b) ⊆ σπ,k,e(c) ⊆

spπ,k,e(c) ⊆⋃

p+q=k

spπ,p,e(a)× spπ,q(b)⋃ ⋃

p+q=k

spπ,p(a)× spπ,q,e(b).

21

Next an idea of the proof of Theorem A is presented. First of all same

preparation is needed.

Consider the Koszul complex of the representation ρ: L → L(X1⊗X2),

(X1⊗X2 ⊗ ∧L, d(ρ)). Observe that the sets of characters of L may be

naturally identified with the cartesian product of the sets of characters of

L1 and L2. Indeed, it is clear that L∗ ∼= L∗1 × L∗2. Moreover, since as Lie

algebra L is the direct sum of L1 and L2, if [., .] denotes the Lie bracket

of L, then the restriction of [., .] to L1 or L2 coincides with the bracket of

L1 or L2 respectively, and for l1 ∈ L1 and l2 ∈ L2, [l1, l2] = 0. Then, the

map

H: L∗ → L∗1 × L∗2, f 7→ (f ◦ ι1, f ◦ ι2)

defines an identification of the characters of L and the cartesian product

of the characters of L1 and L2, where ιj : Lj → L denotes the inclusion

map, j = 1, 2. In the proof of the theorem this identification will be used.

Next consider L1 and L2 two complex solvable finite dimensional Lie

algebras of dimensions n and m respectively, X1 and X2 two complex

Banach spaces, and ρi: Li → L(Xi), i = 1, 2, two representations of the

Lie algebras. Also consider the Koszul complexes associated to the rep-

resentations ρ1 and ρ2, that is, (X1 ⊗ ∧L1, d(ρ1)) and (X2 ⊗ ∧L2, d(ρ2))

respectively.

Suppose that there is a tensor product of X1 and X2 with respect to

〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2. Then, it is easy to prove that there is a

well-defined tensor product of X1⊗∧pL1 and X2⊗∧qL2, X1⊗∧pL1⊗X2⊗∧qL2, relative to 〈X1⊗∧pL1, X1⊗∧pL1

′〉 and 〈X2⊗∧qL2, X2⊗∧qL2′〉, for

all p = 0, . . . , n and q = 0, . . . , m. Furthermore, it is possible to consider

the following diagram

X1 ⊗ ∧p−1L1⊗X2 ⊗ ∧qL2dp(ρ1)⊗Iq←−−−−−− X1 ⊗ ∧pL1⊗X2 ⊗ ∧qL2

(−1)p−1Ip−1⊗dq(ρ2)

y (−1)pIp⊗dq(ρ2)

yX1 ⊗ ∧p−1L1⊗X2 ⊗ ∧q−1L2

dp(ρ1)⊗Iq−1←−−−−−−−− X1 ⊗ ∧pL1⊗X2 ⊗ ∧q−1L2.

Therefore, it is possible to consider the complex of Banach spaces defined

by the tensor product of (X1⊗∧L1, d(ρ1)) and (X2⊗∧L2, d(ρ2)), denoted

by (X1⊗∧L1, d(ρ1))⊗(X2⊗∧L2, d(ρ2)). This complex is the total complex

of the above double complex, that is, for k such that 0 ≤ k ≤ n + m, the

k space is⊕

p+q=k X1 ⊗ ∧pL1⊗X2 ⊗ ∧qL2, and the boundary map, dk,

restricted to X1 ⊗∧pL1⊗X2 ⊗∧qL2 is dk = dp(ρ1)⊗ Iq + (−1)p ⊗ dq(ρ2).

22

On the other hand, consider the direct sum of the Lie algebras L1 and

L2, L = L1×L2, which is a complex solvable finite dimensional Lie algebra,

and the tensor product representation of L in X1⊗X2, i.e.,

ρ = ρ1 × ρ2: L→ L(X1⊗X2), ρ1 × ρ2(l1, l2) = ρ1(l1)⊗ I + I ⊗ ρ2(l2),

where I denotes the identity operator of both X2 and X1. In particular, we

may consider the Koszul complex of the representation ρ: L→ L(X1⊗X2),

i.e., (X1⊗X2 ⊗ ∧L, d(ρ)).

Next consider the map of complex of Banach space Φ = (Φk)0≤k≤n+m:

Φk:⊕p+q=kX1 ⊗ ∧pL1⊗X2 ⊗ ∧qL2 → X1⊗X2 ⊗ ∧kL,

Φ(x1 ⊗ w1⊗x2 ⊗ w2) = x1⊗x2 ⊗ w1 ∧ w2,

where xi ∈ Xi and wi ∈ ∧Li, i = 1, 2. Then a straightforward calculation

shows that Φ is an isomorphism of Banach space complexes between (X1⊗∧L1, d(ρ1))⊗(X2 ⊗ ∧L2, d(ρ2)) and (X1⊗X2 ⊗ ∧L, d(ρ)).

Next the idea of the proof of Theorem A is given.

Consider α ∈ σδ,p(ρ1), β ∈ σδ,q(ρ2), p+q = k, and the Koszul complexes

associated to the representations ρ1 − α: L1 → L(X1) and ρ2 − β: L2 →L(X2), (X1⊗∧L1, d(ρ1−α)) and (X2⊗∧L2, d(ρ2−β)) respectively. Then,

there is p1, 0 ≤ p1 ≤ p, and q2, 0 ≤ q2 ≤ q, such that Hp1(X1⊗∧L1, d(ρ1−α)) 6= 0 and that Hq2(X2 ⊗ ∧L2, d(ρ2 − β)) 6= 0.

Now well, (X1 ⊗ ∧L1, d(ρ1 − α))⊗(X2 ⊗ ∧L2, d(ρ2 − β)) is isomorphic

to (X1⊗X2 ⊗ ∧L, d(ρ− (α, β)).

On the other hand, according to the work of J. Eschmeier (J. reine

angew. Math. 1988), there is a well defined injective linear map

ϕ: H∗(X1⊗∧L1, d(ρ1−α))⊗H∗(X2⊗∧L2, d(ρ2−β))→ H∗(X1⊗X2⊗∧L, d(ρ−(α, β)),

ϕ([a]⊗ [b] = [a⊗ b].

Therefore, according to the structure of the map ϕ, Hp1+q2((X1 ⊗∧L1, d(ρ1 − α))⊗(X2 ⊗ ∧L2, d(ρ2 − β))) 6= 0. However, according to

the above identification of complexes of Banach spaces, Hp1+q2(X1⊗X2⊗∧L, d(ρ − (α, β))) 6= 0. In particular, since 0 ≤ p1 + q2 ≤ p + q = k,

(α, β) ∈ σδ,k(ρ).

23

The middle inclusion is clear.

For the rightmost inclusion, it will be proved that if (α, β) does not be-

long to⋃

p+q=k spδ,p(ρ1)×spδ,q(ρ2), then (α, β) does not belong to spδ,k(ρ).

To this end, a homotopy operator will be cosntructed. There are several

cases to be considered.

First suppose that α /∈ spδ,k(ρ1). Thus, the complex (X1⊗∧L1, d(ρ1−α)) is split for p = 0, . . . , k, i.e., for p = 0, . . . , k there are bounded linear

operators hp: X1 ⊗ ∧pL1 → X1 ⊗ ∧p+1L1, such that hp−1dp(ρ1 − α) +

dp+1(ρ1 − α)hp = Ip, where Ip denotes the identity of X1 ⊗ ∧pL1. Then,

if p and q are such that 0 ≤ p + q ≤ k, define

Hp,q: X1⊗∧pL1⊗X2⊗∧qL2 → X1⊗∧p+1L1⊗X2⊗∧qL2, Hp,q = hp⊗ Iq,

where Iq denotes the identity map of X2 ⊗ ∧qL2. We observe that since

L(X1 ⊗ ∧pL1, X1 ⊗ ∧p+1L1) = L(X1 ⊗ ∧pL1, X1 ⊗ ∧p+1L1), Hp,q is a

well-defined map. Moreover, a direct calculation shows that the maps

Hr, r = 0, . . . , k, Hr =⊕

p+q=r Hp,q, define a homotopy operator for the

complex (X1 ⊗ ∧L1)⊗(X2 ⊗ ∧L2), for r = 0, . . . , k. Thus, according to

the identification of complexes of Banach spaces, the complex (X1⊗X2 ⊗∧L, d(ρ− (α, β))) is split for r = 0, . . . , k, i.e., (α, β) does not belongs to

spδ,k(ρ).

By a similar argument, it is possible to prove that if β /∈ spδ,k(ρ2),

then (α, β) does not belongs to spδ,k(ρ). Thus, we may suppose that

α ∈ spδ,k(ρ1) and β ∈ spδ,k(ρ2).

Now, since (α, β) does not belong to⋃

p+q=k spδ,p(ρ1) × spδ,q(ρ2) and

α ∈ spδ,k(ρ1), we have β /∈ spδ,0(ρ2). Similarly, since β ∈ spδ,k(ρ2) we have

α /∈ spδ,0(ρ1). Thus, there is p1, 1 ≤ p1 ≤ k, such that α /∈ spδ,p1−1(ρ1),

α ∈ spδ,p1 , and β /∈ spδ,k−p1(ρ2).

In order to construct a homotopy operator for the Koszul complex as-

sociated to ρ − (α, β), (α, β) as in the last paragraph, it is necessary to

consider several cases. In fact, the operator will be defined according to

the relation of p and q with p1 and k − p1 respectively, and for each par-

ticular case, it will be proved that the operator is a homotopy. At the end

of the proof, it is clear that this map is a well-defined homotopy for the

Koszul complex of ρ at r = 0, . . . , k.

Moreover, according to the identification of complexes of Banach spaces,

it is enough to prove that the complex (X1 ⊗ ∧L1, d(ρ1 − α))⊗(X2 ⊗

24

∧L2, d(ρ2 − β)) is split in dimension r = 0, . . . , k. Now, the r-space of

this complex is⊕

p+q=k X1 ⊗ ∧pL1⊗X2 ⊗ ∧qL2. The operator Hp,q will

be constructed satisfying the homotopy identity for p and q such that

p + q = r, and then we verify that (Hr)0≤r≤k is a homotopy operator for

the complex, where Hr =⊕

p+q=r Hp,q. The construction of the maps

Hp,q is divided into five cases.

First suppose that 0 ≤ p ≤ p1 − 1 and 0 ≤ q ≤ k − p1. Then, there are

well-defined maps

X1 ⊗ ∧p−1L1hp−1−−−→ X1 ⊗ ∧pL1

hp−→ X1 ⊗ ∧p+1L1,

such that dp+1(ρ1 − α)hp + hp−1dp(ρ1 − α) = Ip, where Ip denotes the

identity of X1 ⊗ ∧pL1, and

X2 ⊗ ∧q−1L2gq−1−−−→ X2 ⊗ ∧qL2

gq−→ X2 ⊗ ∧q+1L2,

such that dq+1(ρ2 − β)gq + gq−1dq(ρ2 − β) = Iq, where Iq denotes the

identity of X2 ⊗ ∧qL2.

Thus, it is possible to define the map

Hp,q: X1⊗∧pL1⊗X2⊗∧qL2 → X1⊗∧p+1L1⊗X2⊗∧qL2⊕X1⊗∧pL1⊗X2⊗∧q+1L2,

Hp,q = 1/2(hp ⊗ Iq ⊕ (−1)pIp ⊗ gq).

Observe that according to the properties of the tensor product, Hp,q is a

well-defined map.

In addition, since p − 1 < p ≤ p1 − 1 and q − 1 < q ≤ k − p1, it is

also possible to define the maps Hp−1,q and Hp,q−1. However, a direct

calculation shows that in X1 ⊗ ∧pL1⊗X2 ⊗ ∧qL2,

dr+1Hp,q + (Hp−1,q ⊕Hp,q−1)dr = I,

where d and I denote the boundary and the identity of the complex (X1⊗∧L1, d(ρ1 − α))⊗(X2 ⊗ ∧L2, d(ρ2 − β)) respectively.

The other cases are when: p ≤ pq − 1 and q = k − p1 − 1; p ≤ p1 − 1

and q > k− p1 − 1; p = p1 and q ≤ k− p1; and p ≥ pq + 1 and q ≤ k− p1.

25

Next the idea of the proof of Theorem B is considered.

The proof of the first part is similar to the correspondign part of The-

orem A.

With regard to the rightmost inclusion, it will be proved by an induction

argument.

First of all the case k = 0 is studied.

Consider a pair (α, β) ∈ spδ,0,e(ρ) \ (spδ,0,e(ρ1)× spδ,0(ρ2)∪ spδ,0(ρ1)×spδ,0,e(ρ2)). Now, since according to Theorem A (α, β) ∈ spδ,0(ρ1) ×spδ,0(ρ2), α ∈ spδ,0(ρ1) \ spδ,0,e(ρ2) and β ∈ spδ,0(ρ2) \ spδ,0,e(ρ2). In

particular, there are bounded linear maps

h0: X1 → X1 ⊗ ∧1L1, g0: X2 → X2 ⊗ ∧1L2,

and finite range projectors

k0: X1 → X1, k′

0: X2 → X2,

such that

d1(ρ1 − α)h0 = I0 − k0, d1(ρ2 − β)g0 = I0 − k′

0.

Now, if H0 is the map

H0: X1⊗X2 → X1⊗X2⊗∧1L2⊕X1⊗∧1L1⊗X2, H0 = (I0⊗ g0, h0⊗ I0),

then it is easy to prove that

d1H0 = I − k0 ⊗ k′

0,

where d and I denote the boundary and the identity of the complex

(X1 ⊗ ∧L1, d(ρ1 − α))⊗(X2 ⊗ ∧L2, d(ρ2 − β)) respectively. However, it

is not difficult to prove that the map k0 ⊗ k′

0 is a projector with finite di-

mensional range. In particular, (α, β) does not belong to spδ,k,e(ρ), which

is impossible according to our assumption.

Next suppose that rightmost inclusion is true for 0 and for all the natural

numbres less than k. If it were not true for k and for (α, β), it is possible

to construct a direct complement to N(dk(ρ − (α, β)) and to prove that

N(dk(ρ− (α, β))/R(dk+1(ρ− (α, β)) is finite dimensional.

26

Next consider an operator ideal between Banach spaces and the multi-

plication representation.

Now, let L1 and L2 be two complex solvable finite dimensional Lie

algebras, X1 and X2 two complex Banach spaces, and ρi: Li → L(Xi), i =

1, 2, two representations of Lie algebras. We consider the Lie algebra Lop2

and the adjoint representation ρ∗2: Lop2 → L(X2

′). Now, if L is the direct

sum of L1 and Lop2 , L = L1 × Lop

2 , then the multiplication representation

of L in J is

ρ: L→ L(J), ρ(l1, l2)(T ) = ρ1(l1)T + Tρ2(l2).

When J is viewed as a tensor product of X1 and X2′ relative to 〈X1, X1

′〉and 〈X2

′, X2〉, ρ coincides with the representation

ρ1 × ρ∗2: L→ L(X1⊗X2′), ρ1 × ρ∗2(l1, l2) = ρ1(l1)⊗ I + I ⊗ ρ∗2(l2).

Moreover, it is easy to prove that the complex (X1⊗∧L1, d(ρ1))⊗(X2⊗∧Lop

2 , d(ρ∗2)) is well defined, and that it is isomorphic to the complex

((X1⊗X2′) ⊗ ∧L, d(ρ1 × ρ∗2)), which may be identified with the complex

(J⊗∧L, d(ρ)), when J is viewed as a tensor product of X1 and X2′ relative

to 〈X1, X1′〉 and 〈X2

′, X2〉.

In the following theorems the joint spectra of the representation ρ are

described.

27

Theorem C. Let L1 and L2 be two complex solvable finite dimensional

Lie algebras, X1 and X2 two complex Banach spaces, and ρi: Li → L(Xi),

i = 1, 2, two representations of Lie algebras. Suppose that there is an

operator ideal J between X2 and X1, and present it as the tensor product of

X1 and X2′ relative to 〈X1, X1

′〉 and 〈X2′, X2〉. Then, if the multiplication

representation ρ: L1 × Lop2 → L(J) is considered,

i)⋃

p+q=k

σδ,p(ρ1)× (σπ,m−q(ρ2)− h2) ⊆ σδ,k(ρ) ⊆

spδ,k(ρ) ⊆⋃

p+q=k

spδ,p(ρ1)× (spπ,m−q(ρ2)− h2),

ii)⋃

p+q=k

σπ,p(ρ1)× (σδ,m−q(ρ2)− h2) ⊆ σπ,k(ρ) ⊆

spπ,k(ρ) ⊆⋃

p+q=k

spπ,p(ρ1)× (spδ,m−q(ρ2)− h2),

where h2 is the character of L2 considered in Theorem 4.

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

Theorem C’. Let X1 and X2 be two complex Banach spaces, and a =

(a1, . . . , an) and b = (b1, . . . , bm) two tuples of operators, ai ∈ L(X1),

1 ≤ i ≤ n, and bj ∈ L(X2), 1 ≤ j ≤ m, such that the vector subspace

generated by them, 〈ai〉1≤i≤n and 〈bj〉1≤j≤m, are nilpotent Lie subalgebras

of L(X1) and L(X2) respectively. Consider J ⊆ L(X2, X1) an operator

ideal between X2 and X1 and the (n+m)-tuple of operators defined in L(J),

c = (La1 , . . . , Lan, Rb1 , . . . , Rbm

), where if S ∈ L(X1) and if T ∈ L(X2),

the maps LS, RT : J → J are defined by

LS(U) = SU, RT (U) = UT.

Then,

i)⋃

p+q=k

σδ,p(a)×σπ,m−q(b) ⊆ σδ,k(c) ⊆ spδ,k(c) ⊆⋃

p+q=k

spδ,p(a)×spπ,m−q(b),

ii)⋃

p+q=k

σπ,p(a)×σδ,m−q(b) ⊆ σπ,k(c) ⊆ spπ,k(c) ⊆⋃

p+q=k

spπ,p(a)×spδ,m−q(b).

28

Theorem D. Let L1 and L2 be two complex solvable finite dimensional

Lie algebras, X1 and X2 two complex Banach spaces, and ρi: Li → L(X),

i = 1, 2, two representations of Lie algebras. Suppose that there is an

operator ideal J between X2 and X1, and we present it as the tensor prod-

uct of X1 and X2′ relative to to 〈X1, X1

′〉 and 〈X2′, X2〉. Then, if the

multiplication representation ρ: L1 × Lop2 → L(J) is considered,

i)⋃

p+q=k

σδ,p,e(ρ1)× (σπ,m−q(ρ2)− h2)⋃ ⋃

p+q=k

σδ,p(ρ1)× (σπ,m−q,e(ρ2)− h2)

⊆ σδ,k,e(ρ) ⊆ spδ,k,e(ρ) ⊆⋃p+q=k

spδ,p,e(ρ1)× (spπ,m−q(ρ2)− h2)⋃ ⋃

p+q=k

spδ,p(ρ1)× (spπ,m−q,e(ρ2)− h2),

ii)⋃

p+q=k

σπ,p,e(ρ1)× (σδ,m−q(ρ2)− h2)⋃ ⋃

p+q=k

σπ,p(ρ1)× (σδ,m−q,e(ρ2)− h2)

⊆ σπ,k,e(ρ) ⊆ spδ,k,e(ρ) ⊆⋃p+q=k

spπ,p,e(ρ1)× (spδ,m−q(ρ2)− h2)⋃ ⋃

p+q=k

spπ,p(ρ1)× (spδ,m−q,e(ρ2)− h2),

where h2 is the character of L2 considered in Theorem 4.

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

Theorem D’. In the conditions of Theorem C’

i)⋃

p+q=k

σδ,p,e(a)× σπ,m−q(b)⋃ ⋃

p+q=k

σδ,p(a)× σπ,m−q,e(b) ⊆ σδ,k,e(c) ⊆

spδ,k,e(c) ⊆⋃

p+q=k

spδ,p,e(a)× spπ,m−q(b)⋃ ⋃

p+q=k

spδ,p(a)× spπ,m−q,e(b),

ii)⋃

p+q=k

σπ,p,e(a)× σδ,m−q(b)⋃ ⋃

p+q=k

σπ,p(a)× σδ,m−q,e(b) ⊆ σπ,k,e(c) ⊆

spπ,k,e(c) ⊆⋃

p+q=k

spπ,p,e(a)× spδ,m−q(b)⋃ ⋃

p+q=k

spπ,p(a)× spδ,m−q,e(b).

29

10. Applications.

Consider two complex Banach spaces X1 and X2, a complex nilpotent

finite dimensional Lie algebra L, and two representations of L, ρ1: L →L(X1) and ρ2: L → L(X2). Suppose that there is a tensor product of X1

and X2 relative to 〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2. Thus, it is possible

to consider the tensor product representation

ρ = ρ1 × ρ2: L× L→ L(X1⊗X2), ρ = ρ1 ⊗ I + I ⊗ ρ2.

Next consider the diagonal map

∆: L→ L× L, ∆(l) = (l, l),

and identify L with ∆(L). In addition, it is possible to consider the rep-

resentation

θ = ρ ◦∆: L→ L(X1⊗X2), θ(l) = ρ1(l)⊗ I + I ⊗ ρ2(l).

In the following theorem the S lodkowski, the split, the essential S lodkowski

and the essential split joint spectra of the representation θ are described.

Theorem E. Let L be a complex nilpotent finite dimensional Lie algebras,

X1 and X2 two complex Banach spaces, and ρi: L→ L(Xi), i = 1, 2, two

representations of the Lie algebra L. We suppose that there is a tensor

product of X1 and X2 relative to 〈X1, X1′〉 and 〈X2, X2

′〉, X1⊗X2. Then,

if we consider the representation θ: L→ L(X1⊗X2), we have

i)⋃

p+q=k

(σδ,p(ρ1)+σδ,q(ρ2)) ⊆ σδ,k(θ) ⊆ spδ,k(θ) ⊆⋃

p+q=k

(spδ,p(ρ1)+spδ,q(ρ2)),

ii)⋃

p+q=k

(σπ,p(ρ1)+σπ,q(ρ2)) ⊆ σπ,k(θ) ⊆ spπ,k(θ) ⊆⋃

p+q=k

(spπ,p(ρ1)+spπ,q(ρ2)),

iii)⋃

p+q=k

(σδ,p,e(ρ1) + σδ,q(ρ2))⋃ ⋃

p+q=k

(σδ,p(ρ1) + σδ,q,e(ρ2)) ⊆ σδ,k,e(θ) ⊆

spδ,k,e(θ) ⊆⋃

p+q=k

(spδ,p,e(ρ1) + spδ,q(ρ2))⋃ ⋃

p+q=k

(spδ,p(ρ1) + spδ,q,e(ρ2)),

iv)⋃

p+q=k

(σπ,p,e(ρ1) + σπ,q(ρ2))⋃ ⋃

p+q=k

(σπ,p(ρ1) + σπ,q,e(ρ2)) ⊆ σπ,k,e(θ) ⊆

spπ,k,e(θ) ⊆⋃

p+q=k

(spπ,p,e(ρ1) + spπ,q(ρ2))⋃ ⋃

p+q=k

(spπ,p(ρ1) + spπ,q,e(ρ2)).

30

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

Next consider two complex Banach spaces X1 and X2, an operator ideal

between X2 and X1, a complex nilpotent Lie algebra L, two representa-

tions of L, ρ1: L → L(X1) and ρ2: L → L(X2), and the representation

of Lop, ν = −ρ2: Lop → L(X2). As before, it is possible to consider the

multiplication representation

ρ: L× L→ L(J), ρ(l1, l2)(T ) = ρ1(l1)T − Tρ2(l2),

and the representation

θ = ρ ◦∆: L→ L(J).

In the following theorem the S lodkowski, the split, the essential S lodkowski

and the essential split joint spectra of the representation θ: L→ L(J) are

described.

31

Theorem F. Let L be a complex nilpotent finite dimensional Lie algebra,

X1 and X2 two complex Banach spaces, and ρi: L→ L(Xi), i = 1, 2, two

representations of the Lie algebra L. We suppose that there is an operator

ideal J between X2 and X1 in the sense of Definition 7. Then, if we

consider the representation θ: L→ L(J), we have

i)⋃

p+q=k

(σδ,p(ρ1)− σπ,m−q(ρ2) + h2) ⊆ σδ,k(θ) ⊆ spδ,k(θ) ⊆

⋃p+q=k

(spδ,p(ρ1)− spπ,m−q(ρ2) + h2),

ii)⋃

p+q=k

(σπ,p(ρ1)− σδ,m−q(ρ2) + h2) ⊆ σπ,k(θ) ⊆ spπ,k(θ) ⊆

⋃p+q=k

(spπ,p(ρ1)− spδ,m−q(ρ2) + h2),

iii)⋃

p+q=k

(σδ,p,e(ρ1)− σπ,m−q(ρ2) + h2)⋃ ⋃

p+q=k

(σδ,p(ρ1)− σπ,m−q,e(ρ2) + h2)

⊆ σδ,k,e(θ) ⊆ spδ,k,e(θ) ⊆⋃p+q=k

(spδ,p,e(ρ1)− spπ,m−q(ρ2) + h2)⋃ ⋃

p+q=k

(spδ,p(ρ1)− spπ,m−q,e(ρ2) + h2),

iv)⋃

p+q=k

(σπ,p,e(ρ1)− σδ,m−q(ρ2) + h2)⋃ ⋃

p+q=k

(σπ,p(ρ1)− σδ,m−q,e(ρ2) + h2)

⊆ σπ,k,e(θ) ⊆ spδ,k,e(θ) ⊆⋃p+q=k

(spπ,p,e(ρ1)− spδ,m−q(ρ2) + h2)⋃ ⋃

p+q=k

(spπ,p(ρ1)− spδ,m−q,e(ρ2) + h2),

where h2 is the character of L2 considered in Theorem 4.

In particular, if X1 and X2 are Hilbert spaces, the above inclusions are

equalities.

32

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