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A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, José Ángel Peláez (2010)

Studia Mathematica

Let μ be a finite positive Borel measure on [0,1). Let μ = ( μ n , k ) n , k 0 be the Hankel matrix with entries μ n , k = [ 0 , 1 ) t n + k d μ ( t ) . The matrix μ induces formally an operator on the space of all analytic functions in the unit disc by the fomula μ ( f ) ( z ) = n = 0 i ( k = 0 μ n , k a k ) z , z ∈ , where f ( z ) = n = 0 a z is an analytic function in . We characterize those positive Borel measures on [0,1) such that μ ( f ) ( z ) = [ 0 , 1 ) f ( t ) / ( 1 - t z ) d μ ( t ) for all f in the Hardy space H¹, and among them we describe those for which μ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².

A model for some analytic Toeplitz operators

K. Rudol (1991)

Studia Mathematica

We present a change of variable method and use it to prove the equivalence to bundle shifts for certain analytic Toeplitz operators on the Banach spaces H p ( G ) ( 1 p < ) . In Section 2 we see this approach applied in the analysis of essential spectra. Some partial results were obtained in [9] in the Hilbert space case.

A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Stephen Bruce Sontz (2013)

Communications in Mathematics

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

A. Perälä, J. A. Virtanen, L. Wolf (2013)

Concrete Operators

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable X and the number of zeros of 1 - Φ ( z ) inside the unit disk, where Φ is the generating function of X , under some mild conditions

Algebras of Toeplitz operators with oscillating symbols.

Albrecht Böttcher, Sergei M. Grudsky, Enrique Ramírez de Arellano (2004)

Revista Matemática Iberoamericana

This paper is devoted to Banach algebras generated by Toeplitz operators with strongly oscillating symbols, that is, with symbols of the form b[eia(x)] where b belongs to some algebra of functions on the unit circle and a is a fixed orientation-preserving homeomorphism of the real line onto itself. We prove the existence of certain interesting homomorphisms and establish conditions for the normal solvability, Fredholmness, and invertibility of operators in these algebras.

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