On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onL _p(⁺,^m) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL ₁(,^mxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

Separation properties for Sturm-Liouville operators

Let q (x) be a positive function given on the interval I of the real axis; let P be the minimal operator generated in L₂(0, +) by the differential expression P[·]=-d²/dx²+q(x); let Q be the operator of multiplication by the function q(x). If D_p* Dq, then P [·] is said to be separated. In this note the separation of P[·] is proved for some growth regularity conditions on the fonction q (x), without assuming anything on its smoothness. One proves that if D_p* D_S, where S is the multiplication operator by the function s(x), satisfying some growth regularity condition, then D_Q D_S.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 349–359, September, 1973.     

Traces of differential functions on subsets of the euclidean space

In the paper one investigates conditions for the existence and for the equality of the traces in the operator sense and in the sense of strict definition for functions from Sobolev and Besov spaces. One gives a complete solution of the problem on the traces and extensions for the trace operator, in the case when is a countably (_m,m)-rectifiable _m-measurable subset of ⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 52–66, 1986.     

On the discriminant function of two commuting nonselfadjoint operators

The notion of the discriminant function of two commuting nonselfadjoint operators A₁ and A₂ in a Hilbert space H was defined by M.S. Livic in a recent paper [5].In the present paper we study the discriminant function for two special cases of commuting operators. In one case A₁ and A₂ are rational functions of one and the same operator A in H. In the second case A₁ and A₂ are Kronecker products of the form A I and I B, where I is the identity operator.As an auxiliary result a relation between the Sylvester and Bezoutian matrices of two polynomials is derived.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

The Schrödinger operator with a weakly accelerating potential

The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/d in L₂() with a one-dimensional Schrödinger operator on the semiaxis with potential v(x), admitting at infinity the estimate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 10–12, 1985.     

Comparing different classes of absolutely summing multilinear operators

Let C denote the composition operator defined on the standard Hardy spaces H^p as where is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C in H^p which are invariant under the shift operator, We restrict our attention to the case where is an inner function. Our main result characterises these invariant subspaces. We also consider C when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.Received: 27 January 2004     

Lp-convergence for expansions in terms of the eigenfunctions of a Sturm — Liouville problem

For the operator L_v=–(x^2ay). x [0, 1], y(0)=y(1)=0 with 0 < 1/2, or ¦y¦ < , y(1)=0 with 1/2 <1, we investigate the effect which the singularity of the Sturm-Liouville operator derived from this self-adjoint expression has on L_p-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in L_p [0, 1] for 2/(2–) < p < 2/.Translated from Matematicheskii Zametki, Vol. 3, No. 6, pp. 683–692, June, 1968.The author is grateful to V. M. Tikhomirov for his many valuable remarks and his constant attention to this work.     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

On the Decomposition of an Operator into a Sum of Four Idempotents

We prove that operators of the form (2 ± 2/n)I + K are decomposable into a sum of four idempotents for integer n > 1 if there exists the decomposition K = K ₁ K ₂ ... K _n, , of a compact operator K. We show that the decomposition of the compact operator 4I + K or the operator K into a sum of four idempotents can exist if K is finite-dimensional. If n trK is a sufficiently large (or sufficiently small) integer and K is finite-dimensional, then the operator (2 – 2/n)I + K [or (2 + 2/n)I + K] is a sum of four idempotents.     

An Operator-valued Berezin Transform and the Class of <Emphasis Type="Italic">n</Emphasis>-Hypercontractions

We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction we associate a positive -valued operator measure dω _{n, T} supported on the closed unit disc in a way that generalizes the above notion of operator-valued Berezin transform. This construction of positive operator measures dω _{n, T} gives a natural functional calculus for the class of n-hypercontractions. We revisit also the operator model theory for the class of n-hypercontractions. The new results here concern certain canonical features of the theory. The operator model theory for the class of n-hypercontractions gives information about the structure of the positive operator measures dω _{n, T} .     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Local properties of accessible injective operator ideals

In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a good behaviour of trace duality, which is canon-ically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of absolutely summing operators (prop. 4.1). By the famous Grothendieck inequality, every operator from L ₁ to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization—via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence, we obtain a class of injective ideals, which are quasi-accessible, and with the help of tensor stability, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.     

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’

We use Voiculescu’s concept of free probability to construct a completely isomorphic embedding of the operator space OH in the predual of a von Neumann algebra. We analyze the properties of this embedding and determine the operator space projection constant of OH_n:

On symmetrizable operators of which some iteration satisfies a positive definite condition

Considered are linear (in general, unbounded) operators A, defined on a set R which is dense in the Hubert Space X, which are symmetrizable by a symmetric operator H in R. Under the condition that there exists an integer p 0 for which (HA^P _x, x) 0 for any x R, the spectral properties of the operator A and the solutions of the equation x - Ax=y, x, y R are investigated. The results obtained are applied to investigating some boundary-value problems for differential equations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 71–76, January, 1969.     

A spectral condition for strong convergence of Markov operators

Summary If P is a conservative Markov operator on L ₁(m) with no unimodular spectral points except 1, then u(I–P)Pⁿ₁0 for every uL ₁.     

Hankel operators and problems of best approximation of unbounded functions

For each function f, f VMO, there exists a unique function f₀, analytic in the circle and such that f–f₀=_f{gVMO_A}. We define the operator of best approximation (nonlinear) A, Af=f₀, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H², then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.     

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.