Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators

1 Introduction

Let B ( X ) d (resp., B ( ℋ ) d ) denote the product of d -copies of the algebra of operators, i.e., bounded linear transformations, on an infinite dimensional complex Banach space X into itself (resp., on an infinite dimensional complex Hilbert space ℋ into itself). A d -tuple A = ( A 1 , … , A d ) ∈ B ( X ) d is a commuting d -tuple if [ A i , A j ] = A i A j − A j A i = 0 for all 1 ≤ i , j ≤ d . Recall from Gleeson and Richter [1] that a commuting d -tuple A ∈ B ( ℋ ) d is m -isometric if

where β = ( β 1 , … , β d ) , β i are non-negative integers for all 1 ≤ i ≤ d , ∣ β ∣ = ∑ i = 1 d β i , β ! = ∏ i = 1 d β i ! , A β = ∏ i = 1 d A i β i , and A * β = ∏ i = 1 d A i * β i . Generalizing this to commuting d -tuple pairs ( A , B ) of Banach space operators, the pair ( A , B ) is said to be m -isometric if

[1–3]. Analogously [3], the commuting d -tuple ( A , B ) is said to be m -symmetric if

The classes consisting of m -isometric and m -symmetric commuting d -tuples have been considered by a number of authors in the recent past (see [1–6] for further references), and it is known that these classes share a number of, though not all, properties with their single operator (i.e., 1-tuple) kin. For example, if A , B , S and T are commuting d -tuples in B ( X ) d , A commutes with S , B commutes with T , and △ A , B m ( I ) = △ S , T n ( I ) = 0 (resp., δ A , B m ( I ) = δ S , T n ( I ) = 0 ) for some positive integers m and n , then △ A S , B T m + n − 1 ( I ) = 0 (resp., δ A S , B T m + n − 1 ( I ) = 0 ); see [3, Corollary 4.2] for the general case and [7] for the 1-tuple case. The converse fails, i.e., given commuting d -tuples A , B , S , and T such that A commutes with S , B commutes with T , and △ A S , B T m + n − 1 ( I ) = 0 (resp., δ A S , B T m + n − 1 ( I ) = 0 ) it does not follow that A , B and S , T (or, scalar multiples thereof) satisfy △ A , B m ( I ) = △ S , T n ( I ) = 0 (resp., δ A , B m ( I ) = δ S , T n ( I ) = 0 ). A case where there is a positive answer is that of the tensor product pairs ( A ⊗ S , B ⊗ T ) : if A , B , S , T are commuting d -tuples such that ( A ⊗ S , B ⊗ T ) is m -isometric (resp., m -symmetric), then there exist integers m i > 0 and a non-zero scalar c such that m = m 1 + m 2 − 1 , ( A , c B ) is strict m 1 -isometric, and ( S , 1 c T ) is strict m 2 -isometric (resp., ( A , c B ) is strict m 1 -symmetric, and ( S , 1 c T ) is strict m 2 -symmetric). Recall here that the pair ( A , B ) is strict m -isometric (resp., strict m -symmetric) if △ A , B m ( I ) = 0 and △ A , B m − 1 ( I ) ≠ 0 (resp., δ A , B m ( I ) = 0 and δ A , B m − 1 ( I ) ≠ 0 ). This article considers tensor sum pairs ( S ⊗ I + I ⊗ Q , T ⊗ I ) , S , T , and Q commuting d -tuples in B ( X ) d , and ( T 1 * ⊗ I + I ⊗ T 2 * , T 1 ⊗ I + I ⊗ T 2 ) , T 1 and T 2 commuting d -tuples in B ( ℋ ) d . We prove that if ( S ⊗ I + I ⊗ Q , T ⊗ I ) is strictly m -isometric, then there exist positive integers m i and a scalar d -tuple λ = ( λ 1 , … , λ d ) such that m = m 1 + m 2 − 1 , ( S + λ , T ) is m 1 -isometric and Q − λ is m 2 -nilpotent; if ( T 1 * ⊗ I + I ⊗ T 2 * , T 1 ⊗ I + I ⊗ T 2 ) is strict m -symmetric (resp., strict m -isometric), then there exist positive integers m i and a scalar d -tuple λ = ( λ 1 , … , λ d ) such that m = m 1 + m 2 − 1 , ( T 1 * + λ ¯ , T 1 + λ ) is m 1 -symmetric and ( T 2 * − λ ¯ , T 2 − λ ) is m 2 -symmetric (resp., m = m 1 + 2 m 2 − 2 , ( T 1 * + λ ¯ , T 1 + λ ) is m 1 -isometric and T 2 − λ is m 2 -nilpotent). This generalizes [8, Theorem 22] and [9, Theorem 1.3] to the case of commuting d -tuples of operators. The argument that we use to prove these results is, in spirit, similar to the argument used in proving related results in [5,7] and depends upon reducing the d -tuple problem to a 1-tuple case.

The plan of the article is as follows. We introduce all relevant terminology, notation, and complementary results in Section 2. The main results are proved in Section 3, and the final section lists the references.

2 Preliminaries

Given operators A , B ∈ B ( X ) , let L A and R B ∈ B ( B ( X ) ) denote, respectively, the operators L A ( X ) = A X and R B ( X ) = X B of left multiplication by A and right multiplication by B . For commuting d -tuples A = ( A 1 , … , A d ) , B = ( B 1 , … , B d ) in B ( X ) d and a scalar d -tuple α = ( α 1 , … , α d ) such that α i is a non-negative integer for all 1 ≤ i ≤ d , let L A α , R B α , the convolution ( L A * R B ) j ( X ) and the multiplication ( L A × R B ) ( X ) , X ∈ B ( X ) , be defined by

Define the operator ∑ i = 1 d A i X B i n by

(Thus, ⌊ A ⌋ n = A ⌊ A ⌋ n − 1 I = I ⌊ A ⌋ n − 1 A = … = A n , ⌊ A B ⌋ n = A ⌊ A B ⌋ n − 1 B = … = A n B n , and ∑ i = 1 d A i B i n = ∑ i = 1 d A i ∑ i = 1 d A i B i n − 1 B i .) The commuting d -tuples A and B commute, [ A , B ] = 0 , if

It is clear that

and if [ A , B ] = 0 , then

A pair ( A , B ) of commuting d -tuples of B ( X ) d operators is m -isometric if

and the pair ( A , B ) is m -symmetric if

Commuting tuples of m -isometric, similarly n -symmetric, operators are known to share a large number of properties with their single operator counterparts: thus, just as for the 1-tuple case, △ A , B m ( I ) = 0 implies △ A , B t ( I ) = 0 , similarly δ A , B m ( I ) = 0 implies δ A , B t ( I ) = 0 , for integers t ≥ m . However, there are instances where a property holds for the single operator version but fails for the d -tuple version: for example, whereas

(similarly, for n -symmetric ( A , B ) ), these properties fail for commuting d -tuples (see [3]).

If ( A , B ) ∈ ( X , m ) -isometric (i.e., △ A , B m ( X ) = ( I − L A ⁎ R B ) m ( X ) = 0 , X ∈ B ( X ) ), then

and if ( A , B ) ∈ ( X , n ) -symmetric (i.e., δ A , B n ( X ) = ( L A − R B ) n ( X ) = 0 , X ∈ B ( X ) ), then

for all integers t ≥ 0 , where

and

Let X ⊗ X denote the completion, endowed with a reasonable cross norm, of the algebraic tensor product of X with X has overline. Let S ⊗ T denote the tensor product of S , T ∈ B ( X ) . The tensor product of the d -tuples A = ( A 1 , … , A d ) and B = ( B 1 , … , B d ) is the d 2 -tuple

(Thus, A ⊗ I = ( A 1 ⊗ I , … , A d ⊗ I ) .) A commuting d -tuple Q = ( Q 1 , … , Q d ) is n -nilpotent for some positive integer n if Q n = ∏ i = 1 d Q i α i = 0 for all d -tuples α = ( α i , … , α d ) of non-negative integers α i such that ∣ α ∣ = ∑ i = 1 d α i = n and Q α is non-zero for at least one α with ∣ α ∣ ≤ n − 1 . Clearly, Q is n -nilpotent if and only if the tensor product operators Q ⊗ I and I ⊗ Q are n -nilpotent. Furthermore, ( A , B ) is m -isometric (resp., m -symmetric) if and only if ( A ⊗ I , B ⊗ I ) and ( I ⊗ A , I ⊗ B ) are m -isometric (resp., m -symmetric).

Given a commuting d -tuple A = ( A 1 , … , A d ) and a scalar λ = ( λ 1 , … , λ d ) , we abbreviate ( A − λ I ) = ( A 1 − λ 1 I , … , A d − λ d I ) to A − λ .

3 Results

We start with a technical lemma. Recall that a pair ( A , B ) of commuting d -tuples in B ( X ) d is strict m -isometric if △ A , B m ( I ) = 0 and △ A , B m − 1 ( I ) ≠ 0 . Since △ A , B m ( I ) = 0 implies △ A , B t ( I ) = 0 for integers t ≥ m , ( A , B ) strict m -isometric implies the linear independence of the sequence { △ A , B r ( I ) } r = 0 m − 1 . A similar statement holds for strict n -symmetric operators, i.e., if δ A , B n ( I ) = 0 and δ A , B n − 1 ( I ) ≠ 0 , then the sequence { δ A , B r ( I ) } r = 0 n − 1 is linearly independent.

If, for commuting d -tuples S , T , and Q , the pair ( S , T ) is m 1 -isometric and Q is m 2 -nilpotent, then ( S ⊗ I , T ⊗ I ) is m 1 -isometric, I ⊗ Q is m 2 -nilpotent, and ( S ⊗ I + I ⊗ Q , T ⊗ I ) is ( m 1 + m 2 − 1 ) -isometric [3]. The converse has a solution, as the following theorem, proved for the single linear operators case in [8] (see also [9]), proves.