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The essential spectrum of holomorphic Toeplitz operators on H p spaces

Mats Andersson, Sebastian Sandberg (2003)

Studia Mathematica

We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators T g on H p ( D ) , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of T g provided that g - 1 ( 0 ) is a compact subset of D.

The joint essential numerical range, compact perturbations, and the Olsen problem

Vladimír Müller (2010)

Studia Mathematica

Let T₁,...,Tₙ be bounded linear operators on a complex Hilbert space H. Then there are compact operators K₁,...,Kₙ ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T₁-K₁,...,Tₙ-Kₙ) equals the joint essential numerical range of (T₁,...,Tₙ). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ ℕ then there exists a compact operator K ∈ B(H) such that | | ( S - K ) | | = | | S | | e . This generalizes results of C. L. Olsen.

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li, Yiu-Tung Poon (2009)

Studia Mathematica

Let W(A) and W e ( A ) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that W e ( A ) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, W e ( A ) can be obtained as the intersection of all sets of the form c l ( W ( A , . . . , A i + 1 , A i + F , A i + 1 , . . . , A ) ) , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in W e ( A ) as star centers....

The Słodkowski spectra and higher Shilov boundaries

Vladimír Müller (1993)

Studia Mathematica

We investigate relations between the spectra defined by Słodkowski [14] and higher Shilov boundaries of the Taylor spectrum. The results generalize the well-known relation between the approximate point spectrum and the usual Shilov boundary.

(Ultra)differentiable functional calculus and current extension of the resolvent mapping

Mats Andersson (2003)

Annales de l’institut Fourier

Let a = ( a 1 , ... , a n ) be a tuple of commuting operators on a Banach space X . We discuss various conditions equivalent to that the holomorphic (Taylor) functional calculus has an extension to the real-analytic functions or various ultradifferentiable classes. In particular, we discuss the possible existence of a functional calculus for smooth functions. We relate the existence of a possible extension to existence of a certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of a . If a ...

Vasilescu-Martinelli formula for operators in Banach spaces

V. Kordula, V. Müller (1995)

Studia Mathematica

We prove a formula for the Taylor functional calculus for functions analytic in a neighbourhood of the splitting spectrum of an n-tuple of commuting Banach space operators. This generalizes the formula of Vasilescu for Hilbert space operators and is closely related to a recent result of D. W. Albrecht.

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