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). Integral Equations and Operator Theory Springer Journals

The Essential Spectrum of Toeplitz Operators on the Unit Ball

Loading next page...
Springer International Publishing
Copyright © 2017 by Springer International Publishing AG
Mathematics; Analysis
Publisher site
See Article on Publisher Site


You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial