Convexity properties of operator radii associated with unitary $\rho$-dilations.

Andô, Tsuyoshi; Nishio, Katsuyoshi

doi:10.1307/mmj/1029001147

Cited by 9 publications

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“…Remark. This corollary gives a different approach to a result of Ando and K. Nishio (see [3] or [2, Theorem 3.5]), who showed that logw p (T) is convex for all p Ç (0, oo ).…”

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confidence: 93%

Unitarily Invariant Operator Norms

Fong

Holbrook

1983

Can. j. math.

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1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius1of a Hilbert space operator T. Among the many interesting developments, we may mention:(a) C. Berger's proof of the “power inequality”2(b) R. Bouldin's result that3for any isometry V commuting with T;(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

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“…Remark. This corollary gives a different approach to a result of Ando and K. Nishio (see [3] or [2, Theorem 3.5]), who showed that logw p (T) is convex for all p Ç (0, oo ).…”

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confidence: 93%

Unitarily Invariant Operator Norms

Fong

Holbrook

1983

Can. j. math.

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“…A. R. Holbrook [15], properties (xiii)-(xvi) were discovered by T. Ando and K. Nishio [4]. Property (xix) was shown by K. Okubo and T. Ando [26], and follows also from (xvi) and (xviii).…”

Section: Vol 99 (9999)mentioning

confidence: 89%

Multivariable ρ-contractions

Kalyuzhnyĭ-Verbovetzkiĭ

2005

Recent Advances in Operator Theory and Its Applications

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Abstract. We suggest a new version of the notion of ρ-dilation (ρ > 0) of an N -tuple A = (A1, . . . , AN ) of bounded linear operators on a common Hilbert space. We say that A belongs to the class Cρ,N if A admits a ρ-dilation A = ( A1, . . . , AN ) for which ζ A := ζ1 A1 + · · · + ζN AN is a unitary operator for each ζ := (ζ1, . . . , ζN ) in the unit torus T N . For N = 1 this class coincides with the class Cρ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of Cρ,1 = Cρ to the case of Cρ,N , N > 1, using so-called Agler kernels. Also, the notion of operator radii wρ, ρ > 0, is generalized to the case of N -tuples of operators, and to the case of bounded (in a certain strong sense) holomorphic operator-valued functions in the open unit polydisk D N , with preservation of all the most important their properties. Finally, we show that for each ρ > 1 and N > 1 there exists an A = (A1, . . . , AN ) ∈ Cρ,N which is not simultaneously similar to any T = (T1, . . . , TN ) ∈ C1,N , however if A ∈ Cρ,N admits a uniform unitary ρ-dilation then A is simultaneously similar to some T ∈ C1,N . Mathematics Subject Classification (2000). Primary 47A13; Secondary 47A20, 47A56.

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“…Now, we define the function g on [1, +∞[ by setting g(ρ) = ρw ρ (S n+1 ). Suppose that there exist ρ 1 , ρ 2 in ]1, +∞[ with ρ 1 < ρ 2 and such that g(ρ 1 ) = g(ρ 2 ), then using the facts that s → h(s) = g(1 + e s ) is a convex function on the real line (see Theorem 4 of [1]), and that lim s→−∞ h(s) = S n+1 = 1, we derive that h is increasing on R. Taking into account that g(ρ 1 ) = g(ρ 2 ), we see that h must be constant on ]−∞, ln(ρ 2 − 1)]. Thus we should have 1 = g(ρ 1 ) = g(ρ 2 ) which is impossible since 1 < ρw ρ (S n+1 ) for any ρ > 1 (ρ 2 − a 2 = D 1 (a) > 0).…”

Section: Some Consequencesmentioning

confidence: 99%

Harnack parts for some truncated shifts

Cassier

Benharrat

2020

Linear and Multilinear Algebra

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The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the ρ-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given. * Corresponding author. 2010 Mathematics Subject Classification. Primary 47A12, 47A20, 47A65; Secondary 15A60.

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Convexity properties of operator radii associated with unitary $\rho$-dilations.

Cited by 9 publications

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Unitarily Invariant Operator Norms

Unitarily Invariant Operator Norms

Multivariable ρ-contractions

Harnack parts for some truncated shifts

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