On Weyl and Browder Spectra of Tensor Products

Abstract

Abstract. Let A and B be Hilbert space operators. In this paper we explore the structure of parts of the spectrum of the tensor product A ⊗ B, and conclude some properties that follow from such a structure. We give conditions on A and B ensuring that, where σ ( · ) and σ w ( · ) stand for the spectrum and Weyl spectrum, respectively. We also investigate the problem of transferring Weyl and Browder's theorems from A and B to their tensor product A ⊗ B.

Weyl Spectral Identity and Biquasitriangularity

“…Is it true that if A and B are isoloid, then the WSI holds? Again, the very same setup discussed in § 4 offers a negative answer to this question, too: the pair of operators that satisfy Weyl's theorem but whose tensor product does not satisfy Browder's theorem, given in [14, § 3], are isoloid, and the WSI does not hold for their tensor product according to [19,Corollary 6] (see Remark 4.2 (b)). Such a pair of operators exhibited in [14, § 3] is constructed as follows.…”

“…The operators A and B satisfy Weyl's theorem, while their tensor product A ⊗ B does not satisfy Browder's theorem [14, § 3]. Hence, according to [19,Corollary 6] (see Remark 4.2 (b)), the WSI does not hold. However, observe that the isolated points 0 and 1 of σ(A) and −1 and 0 of σ(B), are eigenvalues of A and B, and so A and B are isoloid.…”

On Weyl's Theorem for Tensor Products

“…That is, it was not known if there existed a pair of operators A and B for which the above inclusion was proper. This question was solved quite recently by using a counterexample from [4, Section 3] (which exhibits a pair of operators that satisfy Weyl's theorem whose tensor product does not satisfy Browder's theorem) together with Corollary 6 from [7] (which says that Browder's theorem is transferred from a pair of operators to their tensor product if and only if the above inclusion is an identity). Thus, there exists a pair of operators for which the above inclusion is proper.…”

“…and it was also shown that the above inclusion may be proper even if the Weyl spectrum identity holds for A and B, with A, B and A ⊗ B being isoloid operators that satisfy Weyl's theorem (see [7,Remark 2]). …”

The Drazin spectrum of tensor product of Banach algebra elements and elementary operators

“…It is worth noticing that the Drazin spectrum is a key notion in the research area of Weyl's and Browder's theorems and their generalizations. What is more, in the recent past (generalized) Weyl's and (generalized) Browder's theorems of both tensor product operators and elementary operators have been studied ( [23,1,19,8,9,4,13,10]). On the other hand, the set of isolated points of the spectrum is another central notion in this area of research.…”