Operator fractional-linear transformations: convexity and compactness of image; applications

Khatskevich, V. A.; Shulman, Victor S.

doi:10.4064/sm-116-2-189-195

Cited by 18 publications

(11 citation statements)

References 3 publications

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“…In recent publications these functions and their extensions to several variables found important applications to composition operators on functional Hardy and Bergman spaces ( [1] and [5]- [7]), to the Koenigs embedding problem and related Abel-Schröder type functional equations ( [1] and [19]- [23]), to the study of dichotomous behavior of solutions of differential equations in Hilbert space ( [17] and [24]- [27]), and to generating theory of one-parameter semigroups ( [18] and [39]). The infinite-dimensional, operator, analogues of linear fractional functions are linear fractional transformations (LFT) introduced by M. G. Krein ([28] and [29]) for the study of invariant subspaces and spectra of operators on spaces with an indefinite metric.…”

Section: Introductionmentioning

confidence: 99%

Linear fractional relations for Hilbert space operators

Khatskevich

Ostrovskii

Shulman

2006

Mathematische Nachrichten

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Key wordsIn this paper we study linear fractional relations defined in the following way.Let Hi and H i , i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from. To each such operator there corresponds a 2 × 2 operator matrix of the formwhere The map GT is called a linear fractional relation.The main result of the paper is the description of operator matrices of the form ( * ) for which the relation GT is defined on some open ball of the space L (H1, H2).Linear fractional relations are natural generalizations of linear fractional transformations studied by M. G. Krein and Yu. L.Šmuljan (1967).The study of both linear fractional transformations and linear fractional relations is motivated by the theory of spaces with an indefinite metric and its applications.

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Section: Introductionmentioning

confidence: 99%

Linear fractional relations for Hilbert space operators

Khatskevich

Ostrovskii

Shulman

2006

Mathematische Nachrichten

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“…To show that G T is non-constant, assume the contrary. By Proposition 1 this assumption implies that there exists W : B 1 → B 2 such that T 21 = WT 11 and T 22 = WT 12 . From the definitions of T ij we immediately get:…”

Section: Lemma 3 G T Is a Non-constant Everywhere Defined Bounded mentioning

confidence: 99%

“…Another direction of generalization of the classical results is to consider the Banach space case (see [1, pp. 448-449], [3], [4], [9], [10], [11], and references therein). In this paper we combine these directions.…”

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confidence: 99%

Linear Fractional Relations in Banach Spaces: Interior Points in the Domain and Analogues of the Liouville Theorem

Ostrovskii

2007

Glasgow Math. J.

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Abstract. In this paper we study linear fractional relations defined in the following way. Let B i , B i , i = 1, 2, be Banach spaces. We denote the space of bounded linear operators by L. Let T ∈ L(B 1 ⊕ B 2 , B 1 ⊕ B 2 ). To each such operator there corresponds a 2 × 2 operator matrix of the formwhereThe map G T is called a linear fractional relation. The paper is devoted to the following two problems.• Characterization of operator matrices of the form (*) for which the set G T (K) is non-empty for each K in some open ball of the space L(B 1 , B 2 ).• Characterizations of quadruples (B 1 , B 2 , B 1 , B 2 ) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem "a bounded entire function is constant".2000 Mathematics Subject Classification. 47A56, 46B20, 47B50. Introduction.Consider a bounded linear operator T between Banach spaces B, B which can be decomposed into direct sums B = B 1 ⊕ B 2 , B = B 1 ⊕ B 2 . Such a linear operator can be represented by a 2 × 2 operator matrix of the form

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“…On the other hand, many problems from different areas (extensions of operators [16], indefinite metric spaces [12], linear fractional relations [10], operator The second named author was supported by St. John's University Summer 2006 Support of Research Program. The authors thank the referee for helping them to make this paper more readable and error-free.…”

Section: Introductionmentioning

confidence: 99%