多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.     

Invariant subspaces and extremum problems in spaces of vector-valued functions

In this article we obtain, for 1 ? p ? ∞, a characterization of the invariant subspaces of spaces of vector-valued L^p functions defined on the unit circle—i.e., of those subspaces invariant under multiplication by e^ix. This result is then applied to extend, to the corresponding Hardy classes of vector-valued functions, the known characterizations of the extreme points of the unit ball in the scalar Hardy classes H¹ and H^∞. Finally, it is shown that the characterization of the closure of the set of extreme points of the unit ball in H¹ changes significantly when we pass from the scalar to the vector case.     

Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

     

Invariant subspaces of a lightguide with periodic boundary

A waveguide operator is defined. It is proved that its spectrum coincides with the spectrum of a lightguide. The classification of singular points of the continuous spectrum is given. Invariant subspaces of the waveguide operator are distinguished that are related to an interval of the continuous spectrum without singular points. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 51–62.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index

In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate non-trivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.     

A boundary value problem associated to non-homogeneous operators in Orlicz-Sobolev spaces

We apply a three critical points theorem of B. Ricceri to establish the existence of at least three weak solutions for a class of non-homogeneous Neumann problems. Furthermore, by using another theorem of him, we prove that an appropriate oscillating behaviour of the nonlinear term ensures the existence of infinitely many weak solutions. Our analysis is based on recent variational methods for smooth functionals defined on Orlicz-Sobolev spaces.     

A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

We study the solvability of a semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces H_l,p,q, 1<p<∞, depending on a complex parameter q. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Integral boundary value problem for impulsive integro-differential equations in Banach spaces

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.     

Invariant subspaces of some function spaces on a locally compact group

Let G be a σ-compact and locally compact group. If f?L^∞(G) let U_f be the closed subspace of L^∞(G) generated by the left translations of f. Conditions are given which ensure that each function in U_f may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case U_f contains subspaces which are isometrically isomorphic to l¹. If G is metrizable and nondiscrete there is a continuum Γ in L^∞(G) such that, for each f?Γ, U_f contains no non-zero continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, U_f contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. The subspaces U_f above are translation invariant but are not convolution invariant.     

Invariant subspaces for positive operators on locally convex solid Riesz spaces

We prove that an x₀-quasinilpotent semigroup S of continuous positive linear operators on a locally convex solid Riesz space X has a common invariant subspace. Using this, a result which implies the main theorem of Abramovich, Aliprantis and Burkinshaw [J. Funct. Anal. 115 (1993) 418424] is also given.     

A joint similarity problem on vector-valued Bergman spaces

We investigate pairs of commuting Foias-Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting pair of such operators is jointly polynomially bounded if and only if it is similar to a pair of contractions, if and only if both operators are polynomially bounded.     

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

We establish a general identity between the Mahler measures \(\mathrm {m}(Q_k(x,y))\) and \(\mathrm {m}(P_k(x,y))\) of two polynomial families, where \(Q_k(x,y)=0\) and \(P_k(x,y)=0\) are generically hyperelliptic and elliptic curves, respectively.     

On the hilbert boundary value problem for holomorphic function in sobolev spaces

The holomorphic solution φ of the Hilbert boundary value problem Re (a + ib) φ = g on γ in a disk D bounded by the simple closed curve γ has been solved in the space of Hölder-continuously differentiable functions C₁, C_1,α (D) by many authors. Under the assumptionthat g belongs to the Slobodecky space W_s,p (γ), s = 1 ? 1/p,1 < p < ∞ it is shown here that the problem has a uniquesolution in the Sobolev space W_1,p: (D). An a-priori estimate for the norm of in W_{1 p}(D)is given.     

Invariant subspaces and unicellularity of operators of generalized integration in spaces of analytic functionals

Invariant subspaces are described and the unicellularity is proved of one class of operators of generalized integration in spaces of analytic functionals. As one of the realizations it is established that every nontrivial subspace, invariant relative to the integrationF(t)dt, in the space of functions analytic in an arbitrary convex domain (a), is determined by a positive integer m and consists of all functions equal to zero at pointa together with all derivatives up to order m–1.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 221–230, August, 1997.