Compact Hankel operators with conjugate analytic symbols

In 1986 S. Axler [3] proved that forf∈L _a ² the Hankel operator\(H_{\bar f} :L_a^2 \to (L^2 )^ \bot \) is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for\(H_{\bar f} :L_a^p \to L^p \), 1<p<∞. Moreover we prove that\(H_{\bar f} :L_a^1 \to L^1 \) is ?-compact if and only if\(|f'(z)|(1 - |z|^2 )\log \tfrac{1}{{1 - |z|^2 }} \to 0\) as |z|→1^?.     

Hankel operators with unbounded symbols

We prove that there are holomorphic functions in the Hardy space of the unit ball or the bidisc such that the big Hankel operator with symbol is bounded and for any holomorphic function the function cannot be bounded.

     

Compact products of Hankel operators

In this paper we completely characterize compact products of three Hankel operators on the Hardy space. We obtain a necessary and sufficient condition for that , and are simultaneously compact.This work was partly supported by NSF grants. The second author was also partly supported by the Research Council of Vanderbilt University.     

Hankel operators in analytic spaces

A connection between solutions of an operator equation with Hankel matrices is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 998–1000, July, 1990.     

Hankel operators with antiholomorphic symbols on the Fock space

We consider Hankel operators of the form . Here . We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if 2k$">.

     

Compact Hankel operators on harmonic Bergman spaces

We study Hankel operators on the harmonic Bergman spaceb ²(B), whereB is the open unit ball inR ⁿ,n2. We show that iff is in then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb ^p(B), 1<p<, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL ²(B, dV) ontob ²(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in are compact.     

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

On complete pseudoconvex Reinhardt domains in ?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 ?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 space is infinite dimensional and the Hankel operator \({H_{{{\bar z}_1}{{\bar z}_2}}}\) is Hilbert-Schmidt.     

Generalized inversion of finite rank Hankel and Toeplitz operators with rational matrix symbols

The goal of the paper is a generalized inversion of finite rank Hankel operators and Hankel or Toeplitz operators with block matrices having finitely many rows. To attain it a left coprime fractional factorization of a strictly proper rational matrix function and the Bezout equation are used. Generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization.     

Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols

We derive conditions for compactness of Hankel operators () with bounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω⊆Dk.     

Pseudodifferential operators with real analytic symbols and approximation methods for pseudodifferential equations

This paper introduces some methods (including an approximation method) for investigating pseudodifferential equations and related problems (Cauchy problems, boundary value problems,…) based on the technique of pseudodifferential operators with real analytic symbols.     

Hankel operators that commute with second-order differential operators

Suppose that Γ is a continuous and self-adjoint Hankel operator on L²(0,∞) with kernel ?(x+y) and that with a(0)=0. If a and b are both quadratic, hyperbolic or trigonometric functions, and ? satisfies a suitable form of Gauss's hypergeometric differential equation, or the confluent hypergeometric equation, then ΓL=LΓ. The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane.     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

Compact composition operators on weighted Hilbert spaces of analytic functions

We characterize the compactness of composition operators acting on a large family of Hilbert spaces of analytic functions which lie between Bergman and Dirichlet spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.     

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω⊂Cn. We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n−2, then compactness of the Hankel operator Hβ implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C² we show that the condition on β is necessary and sufficient for compactness of Hβ.