# The Essential Spectrum of Toeplitz Operators on the Unit Ball

The Essential Spectrum of Toeplitz Operators on the Unit Ball In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces $$A^p_{\nu }(\mathbb {B}^n)$$ A ν p ( B n ) , where $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$\mathbb {B}^n \subset \mathbb {C}^n$$ B n ⊂ C n denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of $$\mathbb {B}^n$$ B n . It is well-known that then the corresponding Toeplitz operator $$T_f$$ T f is Fredholm if and only if f has no zeros on the boundary $$\partial \mathbb {B}^n$$ ∂ B n . As a consequence, the essential spectrum of $$T_f$$ T f is given by the boundary values of f. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233, 2013, Indiana Univ Math J 56(5):2185–2232, 2007) and limit operator techniques coming from similar problems on the sequence space $$\ell ^p(\mathbb {Z})$$ ℓ p ( Z ) (Hagger et al. in J Math Anal Appl 437(1):255–291, 2016; Lindner and Seidel in J Funct Anal 267(3):901–917, 2014; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495, 1998 and references therein). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

# The Essential Spectrum of Toeplitz Operators on the Unit Ball

, Volume 89 (4) – Sep 6, 2017
38 pages

/lp/springer_journal/the-essential-spectrum-of-toeplitz-operators-on-the-unit-ball-l9EBlLDpiK
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2399-1
Publisher site
See Article on Publisher Site

### Abstract

In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces $$A^p_{\nu }(\mathbb {B}^n)$$ A ν p ( B n ) , where $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$\mathbb {B}^n \subset \mathbb {C}^n$$ B n ⊂ C n denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of $$\mathbb {B}^n$$ B n . It is well-known that then the corresponding Toeplitz operator $$T_f$$ T f is Fredholm if and only if f has no zeros on the boundary $$\partial \mathbb {B}^n$$ ∂ B n . As a consequence, the essential spectrum of $$T_f$$ T f is given by the boundary values of f. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233, 2013, Indiana Univ Math J 56(5):2185–2232, 2007) and limit operator techniques coming from similar problems on the sequence space $$\ell ^p(\mathbb {Z})$$ ℓ p ( Z ) (Hagger et al. in J Math Anal Appl 437(1):255–291, 2016; Lindner and Seidel in J Funct Anal 267(3):901–917, 2014; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495, 1998 and references therein).

### Journal

Integral Equations and Operator TheorySpringer Journals

Published: Sep 6, 2017

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