On subcorrect and correct pairs of operators in Hilbert spaces

Integral Equations and Operator Theory -     

Fredholm operators on locally compact spaces

Sunto Si definiscono operatori localmente ellittici su varietà topologiche; per ogni coppia di operatori localmente ellittici, coincidenti al di fuori di un compatto, si introduce un indice (analitico) di Fredholm. Queste tecniche permettono di estendere i risultati di D. Sullivan e dell'autore [S-T], [T] alla categoria delle varietà topologiche localmente compatte.

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Dedicato al Professor Enzo Martinelli in segno dell'amicizia di cui mi onora

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Research supported by the N.S.F. Grant N MCS 8102758. This paper was written at State University of New York, Center at Stony Brook (June 1982).     

Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certaina priori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such asn- andd-normality, the (generalized and ordinary) Fredholm property, and appropriatea priori estimates.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 902–912, December, 1998.The author is grateful to V. M. Deundyak for useful discussion of this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01195.     

Approximating eigenfunctions of Fredholm operators in Banach spaces

A singular Fredholm operator A is perturbed by an operator of finite rank to obtain an invertible operator B. Theory previously developed for A and B in Hilbert spaces is extended here to Banach spaces. The operator B^?1 is used to construct independent elements in the null spaces N(A), N(A²),…, N(A^k), for some positive integer k, and a basis for N(A) and N(A²). The theory is used to compute approximations to eigenfunctions and generalized eigenfunctions of integral operators.     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Two-interval Sturm-Liouville operators in modified Hilbert spaces

By modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which the Sturm-Liouville equation is defined, we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive. This contrasts with the usual theory which requires the coupling matrix to have determinant one.     

Polynomial interpolation of operators in Hilbert spaces

An operator polynomial is constructed as the limit of a smoothing polynomial as λ → ∞. Using its interpolation properties, necessary and sufficient conditions for the existence of a solution of the polynomial interpolation problem are found. The set of all interpolating polynomials in a Hilbert space is described. Bibliography:4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 44–54.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Invertibility of abstract dual operators in Hilbert spaces

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 77–83, January, 1991.     

Some results about operators in nested Hilbert spaces

With the use of interpolation methods we obtain some results about the domain of an operator acting on the nested Hilbert space {ℋ_f}_f∈∑ generated by a self-adjoint operatorA and some estimates of the norms of its representatives. Some consequences in the particular case of the scale of Hilbert spaces are discussed.     

Products of EP operators on Hilbert spaces

A Hilbert space operator is called the EP operator if the range of is equal to the range of its adjoint . In this article necessary and sufficient conditions are given for a product of two EP operators with closed ranges to be an EP operator with a closed range. Thus, a generalization of the well-known result of Hartwig and Katz (Linear Algebra Appl. 252 (1997), 339-345) is given.

     

On nonlinear differential equations in Hilbert spaces

In a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains a discussion of Medeiros' results, and provides a motivation for the improved uniqueness results in Section 2