Operator Theory and Arithmetic in 𝐻^{∞}

Bercovici, Hari

doi:10.1090/surv/026

Cited by 144 publications

(87 citation statements)

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“…Combining Theorem 3, Theorem 5, and Corollary 7 we see that the only case to consider is when θ is given by (1). By Theorem 14, it suffices to show that B S(θ) * Next, we return to contraction operators of class C 0 .…”

Section: Theorem 15 B S(θ) = {S(θ)}mentioning

confidence: 96%

Spectral radius algebras and $C_0$ contractions

Petrović

2008

Proc. Amer. Math. Soc.

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Abstract. We consider the spectral radius algebras associated to C 0 contractions. If A is such an operator we show that the spectral radius algebra B A always properly contains the commutant of A.Let H be a complex, separable Hilbert space and let L(H) denote the algebra of all bounded linear operators on This paper can be regarded as a sequel to [4] where an extensive investigation of Jordan blocks and C 0 contractions (to be defined below) was conducted. A prominent role in this study was played by the so-called extended eigenvalues. (A complex number λ is an extended eigenvalue of A if there is a nonzero operator X such that AX = λXA.) As we will see, the presence of an eigenvalue or an extended eigenvalue is sufficient to guarantee that B A = {A} . Unfortunately, not every Jordan block S(θ) has either of these. Nevertheless, we will demonstrate that the inclusion under consideration is proper when A belongs to the class C 0 (Theorems 3, 5, and 18). Our method utilizes the relationship between S(θ) and the shift S as well as the quasisimilarity model for C 0 contractions.We briefly review the relevant facts and notation. A contraction A is completely nonunitary if there is no invariant subspace M for A such that A|M is a unitary operator. A completely nonunitary contraction A is said to be of class C 0 if there exists a nonzero function h ∈ H ∞ such that h(A) = 0. The inner function v such that vH ∞ = {u ∈ H ∞ : u(A) = 0} is the minimal function of A and is denoted by m A . A very important subclass of C 0 contractions are the Jordan blocks. Throughout the paper we will use S to denote the forward unilateral shift of multiplicity 1, and {e n } ∞ n=0 the orthonormal basis such that Se n = e n+1 , n ≥ 0. One knows that S can be viewed as multiplication by z on the Hardy space H 2 .

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Section: Theorem 15 B S(θ) = {S(θ)}mentioning

confidence: 96%

Spectral radius algebras and $C_0$ contractions

Petrović

2008

Proc. Amer. Math. Soc.

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“…Full details about the results described in this section can be found in the aforementioned references as well as [5].…”

Section: Background On the Commutant Lifting Theoremmentioning

confidence: 99%

A spectral commutant lifting theorem

Bercovici¹,

Foiaş²,

Tannenbaum³

1991

Trans. Amer. Math. Soc.

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Abstract.The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.

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“…An absolutely continuous contraction T belongs to the class C 0 (T is a C 0 -contraction, T ∈ C 0 ) if there exists a function ϕ ∈ H ∞ , ϕ ≡ 0, such that ϕ(T ) = O. The C 0 -contractions of a certain special form (see [2,III.4.1] and also §1) are called Jordan operators of class C 0 . Every C 0 -contraction T is quasisimilar (see the definition below) to a unique Jordan operator of class C 0 , which is called the Jordan model of T ; see [2,III.5].…”

Section: Introductionmentioning

confidence: 99%

“…The C 0 -contractions of a certain special form (see [2,III.4.1] and also §1) are called Jordan operators of class C 0 . Every C 0 -contraction T is quasisimilar (see the definition below) to a unique Jordan operator of class C 0 , which is called the Jordan model of T ; see [2,III.5]. A certain property (P ) for C 0 -contractions will be used in what follows.…”

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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Operator Theory and Arithmetic in 𝐻^{∞}

Cited by 144 publications

References 0 publications

Spectral radius algebras and $C_0$ contractions

Spectral radius algebras and $C_0$ contractions

A spectral commutant lifting theorem

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

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