The $\bar{\partial }$-Neumann operator and commutators of the Bergman projection and multiplication operators

Friedrich Haslinger

Displaying similar documents to “The $\bar{\partial}$ -Neumann operator and commutators of the Bergman projection and multiplication operators”

The quasi-canonical solution operator to $\bar{\partial}$ restricted to the Fock-space

Georg Schneider (2005)

Czechoslovak Mathematical Journal

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We consider the solution operator $S ℱ_{μ, (p, q)} \to L^{2} {(μ)}_{(p, q)}$ to the $\bar{\partial}$ -operator restricted to forms with coefficients in $ℱ_{μ} = \{f f is entire and \int_{ℂ^{n}} {| f (z) |}^{2} d μ (z) < \infty\}$ . Here $ℱ_{μ, (p, q)}$ denotes $(p, q)$ -forms with coefficients in $ℱ_{μ}$ , $L^{2} (μ)$ is the corresponding $L^{2}$ -space and $μ$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar{\partial}$ . This solution operator will have the property $S v ⊥ ℱ_{(p, q)} \forall v \in ℱ_{(p, q + 1)}$ . As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness...

Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets

Michael Didas

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Let D ⋐ X denote a relatively compact strictly pseudoconvex open subset of a Stein submanifold X ⊂ ℂⁿ and let H be a separable complex Hilbert space. By a von Neumann n-tuple of class over D we mean a commuting n-tuple of operators T ∈ L(H)ⁿ possessing an isometric and weak* continuous $H^{\infty} (D)$ -functional calculus as well as a ∂D-unitary dilation. The aim of this paper is to present an introduction to the structure theory of von Neumann n-tuples of class over D including the necessary function-...

$L ²_{h}$ -domains of holomorphy and the Bergman kernel

Peter Pflug, Włodzimierz Zwonek (2002)

Studia Mathematica

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We give a characterization of $L ²_{h}$ -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak (2005)

Studia Mathematica

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It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L v_{k}$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E \supset L^{\infty} (Ω)$ defined by weighted-sup seminorms and equipped...

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Mehmet Çelik, Yunus E. Zeytuncu (2017)

Czechoslovak Mathematical Journal

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On complete pseudoconvex Reinhardt domains in $ℂ^{2}$ , we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in $ℂ^{2}$ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman...

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

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Let ℳ be a von Neumann algebra with unit $1_{ℳ}$ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_{t} {(x)}_{t \geq 0}$ the generalized s-numbers of x, defined by $μ_{t} (x)$ = inf||xe||: e is a projection in ℳ i with $τ (1_{ℳ} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ \mapsto \int_{0}^{t} l o g μ_{s} (f (λ)) d s$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

Global boundary regularity for the $\bar{p a r t i a l}$ -equation on $q$ -pseudo-convex domains

Heungju Ahn (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For a bounded domain $D$ of $C^{n}$ , we introduce a notion of « $q$ -pseudoconvexity» of new type and prove that for a given $\bar{\partial}$ -closed $(p, r)$ -form $f$ that is smooth up to the boundary on $D$ , and for $r \geq q$ , there exists a $(p, r - 1)$ -form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\bar{\partial} u = f$

The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials

Tomasz Beberok (2017)

Czechoslovak Mathematical Journal

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We investigate the Bergman kernel function for the intersection of two complex ellipsoids ${(z, w_{1}, w_{2}) \in ℂ^{n + 2} : | z_{1} |^{2} + \dots + | z_{n} |^{2} + | w_{1} |^{q} < 1, | z_{1} |^{2} + \dots + | z_{n} |^{2} + | w_{2} |^{r} < 1} .$ We also compute the kernel function for ${(z_{1}, w_{1}, w_{2}) \in ℂ^{3} : | z_{1} |^{2 / n} + | w_{1} |^{q} < 1, | z_{1} |^{2 / n} + | w_{2} |^{r} < 1}$ and show deflation type identity between these two domains. Moreover in the case that $q = r = 2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem. ...

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

The Bergman projection on weighted spaces: L¹ and Herz spaces

Oscar Blasco, Salvador Pérez-Esteva (2002)

Studia Mathematica

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We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces $K_{p}^{q} (w)$ .

Displaying similar documents to “The ∂ ¯ -Neumann operator and commutators of the Bergman projection and multiplication operators”

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