On convergence of functions of normal operators in strong operator topology

2007 ◽  
Vol 41 (3) ◽  
pp. 245-246 ◽  
Author(s):  
O. E. Tikhonov
Keyword(s):  
Strong Operator Topology ◽  
Strong Operator ◽  
Normal Operators ◽  
Functions Of Normal Operators

Iterated Aluthge transforms of composition operators on H2

2015 ◽  
Vol 26 (10) ◽  
pp. 1550079
Author(s):  
Sungeun Jung ◽  
Yoenha Kim ◽  
Eungil Ko
Keyword(s):  
Composition Operators ◽  
Strong Operator Topology ◽  
Weighted Composition Operators ◽  
Strong Operator ◽  
Normal Operators ◽  
Weighted Composition

In this paper, we study various properties of the iterated Aluthge transforms of the composition operators Cφ and Cσ where φ(z) = az + (1 - a) and [Formula: see text] for 0 < a < 1. We express the iterated Aluthge transforms [Formula: see text] and [Formula: see text] as weighted composition operators with linear fractional symbols. As a corollary, we prove that [Formula: see text] and [Formula: see text] are not quasinormal but binormal. In addition, we show that [Formula: see text] and [Formula: see text] are quasisimilar for all non-negative integers n and m. Finally, we show that [Formula: see text] and [Formula: see text] converge to normal operators in the strong operator topology.

scholarly journals A STRONG OPERATOR TOPOLOGY ADIABATIC THEOREM

2002 ◽  
Vol 14 (06) ◽  
pp. 569-584 ◽  
Author(s):  
ALEXANDER ELGART ◽  
JEFFREY H. SCHENKER
Keyword(s):  
Spectral Data ◽  
Time Scale ◽  
Strong Operator Topology ◽  
Continuous Functions ◽  
Adiabatic Theorem ◽  
Strong Operator ◽  
Intrinsic Time ◽  
Embedded Eigenvalues ◽  
Additive Perturbation ◽  
Spectral Projections

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.

scholarly journals Functions of normal operators under perturbations

2011 ◽  
Vol 226 (6) ◽  
pp. 5216-5251 ◽  
Author(s):  
A.B. Aleksandrov ◽  
V.V. Peller ◽  
D.S. Potapov ◽  
F.A. Sukochev
Keyword(s):  
Normal Operators ◽  
Functions Of Normal Operators

scholarly journals UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

2011 ◽  
Vol 53 (3) ◽  
pp. 583-598 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Strong Operator Topology ◽  
Unconditional Convergence ◽  
Fundamental Result ◽  
Boundedness Theorem ◽  
Strong Operator ◽  
Spaces Of Operators ◽  
Funct Anal ◽  
Nikodym Theorem

AbstractIn this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

A Note on M-Ideals of Compact operators

1989 ◽  
Vol 32 (4) ◽  
pp. 434-440 ◽  
Author(s):  
Chong-Man Cho
Keyword(s):  
Strong Operator Topology ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Strong Operator

AbstractSuppose X and Y are closed subspaces of (ΣXn)p and (ΣYn)q (1 < p ≦ q < ∞, dim Xn < ∞, dimYn < ∞), respectively. If K(X, Y), the space of the compact linear operators from X to Y, is dense in L(X, Y), the space of the bounded linear operators from X to Y, in the strong operator topology, then K(X, Y) is an M-ideal in L(X, Y).

Topological aspects of the projective unitary group

1970 ◽  
Vol 68 (1) ◽  
pp. 57-60 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Projective Space ◽  
Unitary Group ◽  
Principal Bundle ◽  
Strong Operator Topology ◽  
Complex Hilbert Space ◽  
Unitary Representations ◽  
Strong Operator ◽  
Unitary Transformations

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

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