Orthogonality of analytic polynomials: a little step further

AbstractFrom the constellation mentioned in Jones and Njåstad (J. Comput. Appl. Math. 105 (1999) 51–91) we have chosen orthogonality of polynomials and moment problems enriching them with operator theory apparatus. Thus this essay resumes the theme of Szafraniec (J. Comput. Appl. Math. 49 (1993) 255) and culminates in updating it with the results of Stochel and Szafraniec (J. Funct. Anal. 159 (1998) 432)

On matrix integration of matrix polynomials

AbstractThis paper is the result of having read a series of recent papers on the quadrature formula for matrix integrals, which caused a strong want of clarifying the circumstances. For this purpose, we have had to revise orthogonality for matrix polynomials being supported by a desire of using means adequate to needs and at the same time of trying to simplify the set-up

Three paths toward the quantum angle operator

We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin-Klauder approaches. One method pertains to Weyl-Heisenberg integral quantization of the plane viewed as the phase space of the motion on the line. It depends on a family of "weight" functions on the plane. The third method rests upon coherent state quantization of the cylinder viewed as the phase space of the motion on the circle. The construction of these coherent states depends on a family of probability distributions on the line.Comment: 20 page

Framings and dilations

The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\em th\'eor\`eme principal} of B. Sz-Nagy go much beyond it.Comment: The final version will appear in Acta Sci. Math (Szeged

Selfadjoint operators in S-spaces

We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by, where U is a unitary operator in. We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space with ρ(A) ̸= ∅ we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. In addition, we give a simple condition for the existence of invariant subspaces

Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established

A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length

It was at the dawn of the historical developments of quantum mechanics when Schrödinger, Kennard and Darwin proposed an interesting type of Gaussian wave packets, which do not spread out while evolving in time. Originally, these wave packets are the prototypes of the renowned discovery, which are familiar as “coherent states” today. Coherent states are inevitable in the study of almost all areas of modern science, and the rate of progress of the subject is astonishing nowadays. Nonclassical states constitute one of the distinguished branches of coherent states having applications in various subjects including quantum information processing, quantum optics, quantum superselection principles and mathematical physics. On the other hand, the compelling advancements of non-Hermitian systems and related areas have been appealing, which became popular with the seminal paper by Bender and Boettcher in 1998. The subject of non-Hermitian Hamiltonian systems possessing real eigenvalues are exploding day by day and combining with almost all other subjects rapidly, in particular, in the areas of quantum optics, lasers and condensed matter systems, where one finds ample successful experiments for the proposed theory. For this reason, the study of coherent states for non-Hermitian systems have been very important. In this article, we review the recent developments of coherent and nonclassical states for such systems and discuss their applications and usefulness in different contexts of physics. In addition, since the systems considered here originated from the broader context of the study of minimal uncertainty relations, our review is also of interest to the mathematical physics communit

Subnormality in the Quantum Harmonic Oscillator

. This is an invitation to enjoy watching how unbounded subnormality, a rather recent development, gets involved in solving the commutation relation of the quantum harmonic oscillator. 1991 Mathematics Subject Classification. Primary 47B20; Secondary 47B15, 47B47, 81S05. Key words and phrases. normal, formally normal and subnormal operators, quantum harmonic oscillator, CCR, creation and annihilation. The research resulting in this paper was supported by the KBN grant # 2 P03A 04110. 2 FRANCISZEK HUGON SZAFRANIEC Leaving appart physical background the quantum harmonic oscillator can be described, in its simplest case, by the commutation relation S S \Gamma SS = I (1) which is the starting point for Hilbert space considerations. Though it seems that everything has been known about it for long time, we would like to have our say in the matter by inserting subnormality in it. The reason for this is in the fact that the very classical solution of (1) represents the most spectacu..