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Linear fractional relations for Hilbert space operators
Author(s) -
Khatskevich V. A.,
Ostrovskii M. I.,
Shulman V. S.
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310400
Subject(s) - mathematics , linear subspace , hilbert space , linear map , bounded function , linear operators , bounded operator , pure mathematics , operator (biology) , ball (mathematics) , linear space , operator theory , combinatorics , discrete mathematics , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene
In this paper we study linear fractional relations defined in the following way. Let ℋ i and ℋ i ′ , i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from ℋ j to ℋ i ′ by L (ℋ j , ℋ i ′ ). Let T ∈ ℒ(ℋ 1 ⊕ ℋ 2 , ℋ 1 ′ ⊕ ℋ 2 ′ ). To each such operator there corresponds a 2 × 2 operator matrix of the formwhere T ij ∈ ℒ (ℋ j , ℋ i ′ ), i, j = 1, 2. For each such T we define a set‐valued map G T from ℒ(ℋ 1 , ℋ 2 ) into the set of closed affine subspaces of ℒ(ℋ 1 ′ , ℋ 2 ′ ) byG T ( K ) = { K ′ ∈ ℒ(ℋ 1 ′ , ℋ 2 ′ ) : T 21 + T 22 K = K ′ ( T 11 + T 12 K )} .The map G T 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 G T is defined on some open ball of the space ℒ(ℋ 1 , ℋ 2 ). 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. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)