On spectral properties of compact Toeplitz operators on Bergman space\n with logarithmically decaying symbol and applications to banded matrices

2020/06/03 by Mahamet Koïta, Koita, Mahamet, Stanislas Kupin +4
Mathematics · #42C10 #47B35 #Algebraic and Geometric Analysis #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #secondary: 30H20

paper · pdf · doi:10.48550/arxiv.2006.02586

openalex publication_date 2020/06/03 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Let L²(D) be the space of measurable square-summable functions on the unit\ndisk. Let L²_a(D) be the Bergman space, i.e., the (closed) subspace of\nanalytic functions in L²(D). P₊ stays for the orthogonal projection going\nfrom L²(D) to L²_a(D). For a function \φ\∈ L^\∞(D), the\nToeplitz operator T_\φ: L²_a(D)\→ L²_a(D) is defined as T_
varphi\nf=P₊
varphi f,
quad f
in L²_a(D). The main result of this article are\nspectral asymptotics for singular (or eigen-) values of compact Toeplitz\noperators with logarithmically decaying symbols, that is \n
varphi(z)=
varphi₁(e^i
theta)
, (1+
log(1/(1-r)))^-
gamma,
quad
gammagt;0,\n where z=re^i\θ and \φ₁ is a continuous (or piece-wise\ncontinuous) function on the unit circle. The result is applied to the spectral\nanalysis of banded (including Jacobi) matrices.\n

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