scholarly journals THE JOINT SIMILARITY PROBLEM FOR WEIGHTED BERGMAN SHIFTS

2002 ◽  
Vol 45 (1) ◽  
pp. 117-139 ◽  
Author(s):  
Sarah H. Ferguson ◽  
Srdjan Petrovic
Keyword(s):  
Hardy Space ◽  
Primary 47B35 ◽  
Bergman Spaces ◽  
Subject Classification ◽  
Weighted Bergman Spaces ◽  
Space Theory ◽  
Single Operator ◽  
Similarity Problem

AbstractWe solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.AMS 2000 Mathematics subject classification: Primary 47B35; 47B47

Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

1998 ◽  
Vol 50 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Frédéric Symesak
Keyword(s):  
Hardy Space ◽  
Finite Type ◽  
Pseudoconvex Domain ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Sufficient Condition ◽  
Schatten Class ◽  
Strictly Pseudoconvex Domain

AbstractThe aimof this paper is to study small Hankel operators h on the Hardy space or on weighted Bergman spaces,where Ω is a finite type domain in ℂ2 or a strictly pseudoconvex domain in ℂn. We give a sufficient condition on the symbol ƒ so that h belongs to the Schatten class Sp, 1 ≤ p < +∞.

scholarly journals The wandering subspace property and Shimorin’s condition of shift operator on the weighted Bergman spaces

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
Changhui Wu ◽  
Zhijie Wang ◽  
Tao Yu
Keyword(s):  
Hardy Space ◽  
Toeplitz Operators ◽  
Shift Operator ◽  
Bergman Spaces ◽  
Invariant Subspaces ◽  
Weighted Bergman Spaces ◽  
Wandering Subspace ◽  
Formula Type

AbstractIn the present paper, we first study the wandering subspace property of the shift operator on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{n})(n=0,2)$$ L a 2 ( d A n ) ( n = 0 , 2 ) via the spectrum of some Toeplitz operators on the Hardy space $$H^{2}$$ H 2 . Second, we give examples to show that Shimorin’s condition for the shift operator fails on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{\alpha })(\alpha >0)$$ L a 2 ( d A α ) ( α > 0 ) .

scholarly journals Integration Operator Acting on Hardy and Weighted Bergman Spaces

2007 ◽  
Vol 75 (3) ◽  
pp. 431-446 ◽  
Author(s):  
Jouni Rättyä
Keyword(s):  
Analytic Function ◽  
Hardy Space ◽  
Asymptotic Formula ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Integration Operator ◽  
Complete Characterisation

Questions related to the operator Jg(f)(z):= ∫xof (ζ)g′(ζ) dζ, induced by an analytic function g in the unit disc, are studied. It is shown that a function G is the derivative of a function in the Hardy space Hp if and only if it is of the form G = Fψ′ where F ∈ Hq, ψ ∈ H3 and 1/s = 1/p − 1/q. Moreover, a complete characterisation of when Jg is bounded or compact from one weighted Bergman space into another is established, and an asymptotic formula for the essential norm of Jg, the distance from compact operators in the operator norm, is given. As an immediate consequence it is obtained that if p < 2 + α and α > −1, then any primitive of belongs to where q = ((2 + α) p)/(2 + α − p). For α = −1 this is a sharp result by Hardy and Littlewood on primitives of functions in Hardy space , 0 < p < 1.

scholarly journals Extension of Zhu's solution to Lotto's conjecture on the weighted Bergman spaces

2004 ◽  
Vol 2004 (41) ◽  
pp. 2199-2203
Author(s):  
Abebaw Tadesse
Keyword(s):  
Hardy Space ◽  
Bergman Space ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces

We reformulate Lotto's conjecture on the weighted Bergman spaceAα2setting and extend Zhu's solution (on the Hardy spaceH2) to the spaceAα2.

On Homogeneous Expansions of Mixed Norm Space Functions in the Ball

1993 ◽  
Vol 36 (1) ◽  
pp. 78-86
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Mixed Norm ◽  
Mixed Norm Space ◽  
Norm Space ◽  
Complex Ball

AbstractFor f analytic in the complex ball having the homogeneous expansion conditions for f to be of Hardy space Hp or of weighted Bergman spaces are expressed in terms of lp properties of the sequence {∥Fk∥p}.

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

scholarly journals Eigenvalues of K-invariant Toeplitz Operators on Bounded Symmetric Domains

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Harald Upmeier
Keyword(s):  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Bounded Symmetric Domains

AbstractWe determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains $$D=G/K,$$ D = G / K , for the irreducible K-types indexed by all partitions of length $$r={\mathrm {rank}}(D)$$ r = rank ( D ) .

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