Operator algebras and lattices of invariant subspaces. II

Капустин, В. М.; Lipin, A. V.

doi:10.1007/bf02111296

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…The relationships between contractions and isometries are studied by many authors. We mentioned here [7], [12], [17], [18], [19], [21], [23], [24], [25], [27], [29], [31] (it should be mentioned that in the last conclusion of [31,Theorem 2.7] V must be replaced by…”

Section: Introductionmentioning

confidence: 99%

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On contractions that are quasiaffine transforms of unilateral shifts

Gamał¹

2018

Preprint

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It is known that if T is a contraction of class C10 and I−T * T is of trace class, then T is a quasiaffine transform of a unilateral shift. Also it is known that if the multiplicity of a unilateral shift is infinite, the converse is not true. In this paper the converse for a finite multiplicity is proved: if T is a contraction and T is a quasiaffine transform of a unilateral shift of finite multiplicity, then I − T * T is of trace class. As a consequence we obtain that if a contraction T has finite multiplicity and its characteristic function has an outer left scalar multiple, then I − T * T is of trace class.Also, it is known that if a contraction T on a Hilbert space H is such that b λ (T )x ≥ δ x for every λ ∈ D, x ∈ H, with some δ > 0, and(here b λ is a Blaschke factor and S1 is the trace class of operators), then T is similar to an isometry. In this paper the converse for a finite multiplicity is proved: if T is a contraction and T is similar to an isometry of finite multiplicity, then T satisfies the above conditions.

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Section: Introductionmentioning

confidence: 99%

“…If T and δ satisfy to the conditions of Theorem 4.1, then δ is a left scalar multiple of the characteristic function of T . Proof is contained in[12, Lemma 4.16].…”

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confidence: 99%

On contractions that are quasiaffine transforms of unilateral shifts

Gamał¹

2018

Preprint

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“…where T 0 is of class C 00 and T 1 is of class C 10 ; see [1,II,§4]. ≺ T ≺ J, where the relation ci ≺ can be realized by at most two operators (see [6,9,10]).…”

Section: Introductionmentioning

confidence: 99%

$C\_\{\textperiodcentered 0\}$-contractions: a Jordan model and lattices of invariant subspaces

Gamał

2004

St. Petersburg Math. J.

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Abstract. A subclass C ·0 (C 0 (P ), fin) of the class of C ·0 -contractions is introduced and studied. This subclass is a generalization of the subclass of C ·0 -contractions with finite defect indices, and it includes the C ·0 -contractions T for which dim Ker T * < ∞ and the defect operator (I − T * T ) 1/2 belongs to the Hilbert-Schmidt class. For an operator of class C ·0 (C 0 (P ), fin), a Jordan model is constructed, and it is proved that the lattices of invariant subspaces remain isomorphic under the quasiaffine transformations.

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On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

Domanov

Malamud

2009

Integr. equ. oper. theory

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Let J α k be a real power of the integration operator J k defined on the Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator A k = n j=1 λjJ α k defined on n j=1 W k p [0, 1]. Namely, we describe the commutant {A k } , the double commutant {A k } and the algebra Alg A k . Moreover, we describe the lattices Lat A k and HypLat A k of invariant and hyperinvariant subspaces of A k , respectively. We also calculate the spectral multiplicity µA k of A k and describe the set Cyc A k of its cyclic subspaces. In passing, we present a simple counterexample for the implicationto be valid. (2000). Primary 47A15, 47A16, 47L80; Secondary 47L10. Mathematics Subject Classification

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Operator algebras and lattices of invariant subspaces. II

Cited by 5 publications

References 8 publications

On contractions that are quasiaffine transforms of unilateral shifts

On contractions that are quasiaffine transforms of unilateral shifts

$C\_\{\textperiodcentered 0\}$-contractions: a Jordan model and lattices of invariant subspaces

On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

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