The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

Existence of zerofree functions <Emphasis Type="Italic">N</Emphasis>-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kre?n’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.     

Representability of de Branges Matrices as Blaschke--Potapov Products and Completeness of Some Families of Functions

Criteria for the representability of meromorphic second-order matrix functions J-expanding in the upper half-plane (de Branges matrices) as left, right, and two-sided Blaschke--Potapov products are stated. Results on the spectral structure of operators whose characteristic matrix functions are de Branges matrices are obtained.     

Vekua Integral Operators on Riemann Surfaces

On an arbitrary (in general, noncompact) Riemann surface R, we study integral operators T and ∏ analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy--Riemann equation $\bar \partial f = F$ in the class of functions f exhibiting Λ₀-behavior in the vicinity of the ideal boundary of R.     

Existence of zerofree functions N-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated (N ? \mathbb Z)({N \in\mathbb {Z}}) to a de Branges Pontryagin space. Let a de Branges Pontryagin space P{\mathcal {P}} and N ? \mathbb Z{N \in \mathbb {Z}} be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to P{\mathcal {P}} in terms of Kreĭn’s Q-function associated with the multiplication operator in P{\mathcal {P}} . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.     

Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1, 1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges–Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges–Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of L² (\mathbbR, dn){L^2 (\mathbb{R}, d\nu)} onto a de Branges space of entire functions which acts as multiplication by a measurable function.     

On a class of completely continuous operators in Hilbert spaces

We introduce a class G of completely continuous operators and prove theorems on the spectral structure of these operators. In particular, operators of this class are similar to model operators in de Branges spaces.     

Approximations in the l∞ norm and the generalized inverse

We introduce the concept of a strict l_∞-metric projector, based in the definition of strict approximation, to prove that for each matrix A of order m×n with coefficients in the field R of real numbers there exists a set of operators G: R^m → Rⁿ homogeneous and continuous, but not necessarily linear (strict generalized inverse) such that AGA = A and 6AGy?y6 is minimized for all y, when the norm is the l_∞ norm. We investigate the properties of these operators and prove that there are two distinguished operators A^-1_{∞, β} and A^-1_∞ which are extensions of the generalized inverse introduced by Newman and Odell in the case of a strictly convex norm.     

On the nature of the de Branges Hamiltonian

We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developing the theory of Hilbert spaces of entire functions (we call them Krein-de Branges spaces), de Branges arrived at a certain class of canonical equations of phase dimension 2. He showed that, for any given Krein-de Branges space, there exists a canonical equation of the class indicated that restores a chain of Krein-de Branges spaces imbedded one into another. The Hamiltonians of such canonical equations are called de Branges Hamiltonians. The following question arises: Under what conditions will the Hamiltonian of a certain canonical equation be a de Branges Hamiltonian? The main theorem of the present work, together with Theorem 1 of the paper cited above, gives an answer to this question.     

An extension of Picard's theorem for meromorphic functions of small hyper-order

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if n∈N and three distinct values of a meromorphic function f of hyper-order less than 1/n² have forward invariant pre-images with respect to a fixed branch of the algebraic function τ(z)=z+αn−1z1−1/n+?+α₁z1/n+α₀ with constant coefficients, then f○τ≡f. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images.     

Hermitian indefinite functions and Pontryagin spaces of entire functions

We establish and investigate a connection between hermitian indefinite continuous functions with finitely many negative squares defined on a finite interval and so-called de Branges spaces of entire functions. This enables us to relate to any hermitian indefinite continuous function on the real axis a certain chain of 2×2-matrix valued entire functions, which are in the positive definite case tightly connected with canonical systems of differential equations.     

A fundamental approach to the generalized eigenvalue problem for self-adjoint operators

The generalized eigenvalue problem for an arbitrary self-adjoint operator is solved in a Gelfand triple consisting of three Hilbert spaces. The proof is based on a measure theoretical version of the Sobolev lemma, and the multiplicity theory for self-adjoint operators. As an application necessary and sufficient conditions are mentioned such that a self-adjoint operator in L₂(R) has (generalized) eigenfunctions which are tempered distributions.     

The spectral theory of contractions on Πk spaces

As we know, B.Sz-Nagy and C.Foins studied systematically contractions on Hilbert spaces and developed the harmonic analysis theory of operators on Hilbert spaces. Since 1950s, people paid great attention to the study of contractions on π_k spaces. Only a few results have been obtained until today; in particular, the spectral theory of contractions on π_k Spaces and corresponding harmonic analysis theory have left still unexplored. This paper, as a continuation of [1], [2], [6], in which the authors after discussing some problems such as the negative invariant subspaces and unitary dilations of contractions on complete spaces with indefinite metrics, establish the triangle model of contractions on π_k spaces and furthermore, apply the triangle model to the study of spectral theory of contractions on π_k spaces, which is essential to the harmonic analysis of operators on π_k spaces.     

Invariants of finite groups generated by generalized transvections in the modular case

We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F _q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F _q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces L _ρ ^p(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L _ρ ^p(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.     

Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

We present an extrapolation theory that allows us to obtain, from weighted L^p inequalities on pairs of functions for p fixed and all A_∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A_∞ weights and also modular inequalities with A_∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz-Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.