Hilbert Spaces Of Entire Functions Research Articles

Cauchy–de Branges Spaces, Geometry of Their Reproducing Kernels and Multiplication Operators

Cauchy–de Branges spaces are Hilbert spaces of entire functions defined in terms of Cauchy transforms of discrete measures on the plane and generalizing the classical de Branges theory. We consider extensions of two important properties of de Branges spaces to this, more general, setting. First, we discuss geometric properties (completeness, Riesz bases) of systems of reproducing kernels corresponding to the zeros of certain entire functions associated to the space. In the case of de Branges spaces they correspond to orthogonal bases of reproducing kernels. The second theme of the paper is a characterization of the density of the domain of multiplication by z in Cauchy–de Branges spaces.

Hilbert spaces and low-lying zeros of L-functions

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of L-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization problem in this setting by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels. In an appendix, we study the related problem of determining the sharp embeddings between the Hilbert spaces associated to the five symmetry types and the classical Paley-Wiener space.

On Montgomery’s pair correlation conjecture: A tale of three integrals

Abstract We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery’s function F ⁢ ( α , T ) {F(\alpha,T)} in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery’s pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery’s pair correlation conjecture.

Necessary Condition for the Existence of Unconditional Bases of Reproducing Kernels for Hilbert Spaces of Entire Functions

Necessary Condition for the Existence of Unconditional Bases of Reproducing Kernels for Hilbert Spaces of Entire Functions

On a Sufficient Condition for the Existence of Unconditional Bases of Reproducing Kernels in Hilbert Spaces of Entire Functions

On a Sufficient Condition for the Existence of Unconditional Bases of Reproducing Kernels in Hilbert Spaces of Entire Functions

On a Certain Class of Entire Functions

In this paper, we examine the problem on the description in terms of the Laplace transform of functionals from the conjugate space for the Hilbert space of entire functions of n variables constructed by a convex function in ℂn, which depends on the modules of variables and grows at infinity faster than a‖z‖ for any a > 0.

Sampling in the image of Sobolev Space in $$L^2({\mathbb {R}}, e^{u^2}du)$$ under Schrödinger semigroup $$e^{it\Delta }$$

It is known that the image of a Sobolev space $$W^{m,2}({\mathbb {R}})$$, $$m\in {\mathbb {N}}$$ in $$L^2({\mathbb {R}}, e^{u^2}du)$$ under the Schrodinger semigroup $$e^{it\Delta }$$ is a Hilbert space of entire functions $$\mathcal {HL}^2({\mathbb {C}}, u_t^m(z)dz)$$. In this article, it is shown that a lattice Z in the complex plane to be a sampling for the space $$\mathcal {HL}^2({\mathbb {C}}, u_t^m(z)dz)$$ if and only if the lower density $$D^{-}(Z)> \frac{1}{8t^2\pi }$$.

On Hilbert Spaces of Entire Functions with Unconditional Bases of Reproducing Kernels

We consider an entire function under certain conditions on the distribution of its zeros. We construct a Hilbert space of entire functions which possess unconditional basis of reproducing kernels at zeros of this function. It is proved that some known Hilbert spaces of entire functions with unconditional bases of reproducing kernels are isomorphic (as normalized spaces) to the corresponding spaces constructed by the entire functions generating the bases.

Generating Functions for Bases in Hilbert Spaces of Entire Functions

We prove that unconditional bases in a functional Hilbert space H have a generating function if and only if the space H is stable. Necessary and sufficient conditions for the stability of spaces adjoint to weighted spaces on an interval are obtained.

Spectral theory of rank one perturbations of normal compact operators

We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis problems for such perturbations. Previously, we developed the spectral theory of rank one perturbations in the selfadjoint case. In the present paper we extend and significantly simplify most of known results in the area. We also prove an Ordering Theorem for invariant subspaces with common spectral part. This result is essentially new even for rank one perturbations of compact selfadjoint operators.

Krein-type theorems and ordered structure for Cauchy–de Branges spaces

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given. We also discuss applications to spectral theory of rank one perturbations of normal operators.

Holomorphic Hermite Functions in Segal–Bargmann Spaces

We study systems of holomorphic Hermite functions in the Segal–Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

De Branges spaces and Fock spaces

Relations between two classes of Hilbert spaces of entire functions, de Branges spaces and Fock-type spaces with nonradial weights, are studied. It is shown that any de Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g. weights depending on the imaginary part only) the corresponding de Branges spaces are explicitly described.

Closed range and cyclicity of composition operators on Hilbert spaces of entire functions

ABSTRACTWe discover a criterion for having a closed range and completely characterize the cyclic behavior of composition operators on a class of Hilbert spaces of entire functions in the complex plane .

Composition operators on Hilbert spaces of entire functions

We obtain a complete description of families of continuous and compact composition operators on Hilbert spaces of entire functions. These operators were introduced by K. Chan and J. Shapiro for studying some dynamic properties of translation operators. In contrast to recent papers devoted to the same problems, we make no additional assumptions on the mentioned spaces. We apply a new research approach based on the embedding of a Hilbert space under consideration into some appropriate Banach space with the sup-norm.

Multiplication operators with deficiency indices (p,p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions

A number of recent papers have established connections between reproducing kernel Hilbert spaces H of entire functions, de Branges spaces, sampling formulas and a class of symmetric operators with deficiency indices (1,1). In this paper analogous connections between reproducing kernel Hilbert spaces of entire vector valued functions, de Branges spaces of entire vector valued functions, sampling formulas and symmetric operators with deficiency indices (p,p) are obtained. Enroute, an analog of L. de Branges' characterization of the reproducing kernel Hilbert spaces of entire functions that are now called de Branges spaces is obtained for the p×1 vector valued case. A special class of these de Branges spaces of p×1 vector valued entire functions is identified as a functional model for M. G. Krein's class of entire operators with deficiency indices (p,p).

Composition Operators on Hilbert Spaces of Entire Functions of Several Variables

We study composition operators acting on Hilbert spaces of entire functions in several variables. Depending on the defining weight sequence of the space, different criteria for boundedness and compactness are developed. Our work extends several known results on Fock spaces and other spaces of entire functions.

Composition operators on Hilbert spaces of entire functions with analytic symbols

Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Carswell, MacCluer and Schuster is generalized to the case of Segal–Bargmann spaces of infinite order. Some related questions are also discussed.

On the 2-mode and k-photon quantum Rabi models

By mapping the Hamiltonians of the 2-mode and 2-photon Rabi models to differential operators in suitable Hilbert spaces of entire functions, we prove that the two models possess entire and normalizable wave functions in the Bargmann–Hilbert spaces only if the frequency [Formula: see text] and coupling strength [Formula: see text] satisfy certain constraints. This is in sharp contrast to the quantum Rabi model for which entire wave functions always exist. For model parameters fulfilling the aforesaid constraints, we determine transcendental equations whose roots give the regular energy eigenvalues of the models. Furthermore, we show that for [Formula: see text], the [Formula: see text]-photon Rabi model does not possess wave functions which are elements of the Bargmann–Hilbert space for all non-trivial model parameters. This implies that the [Formula: see text] case is not diagonalizable, unlike its RWA cousin, the [Formula: see text]-photon Jaynes–Cummings model which can be completely diagonalized for all [Formula: see text].

О гильбертовом пространстве целых функций

О гильбертовом пространстве целых функций