Изометрии действительных подпространств самосопряженных операторов в банаховых симметричных идеалах

Finite Dimensional ◽

Adjoint Operators

Let (mathcal C_E, cdot_mathcal C_E) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space mathcal H. Let mathcal C_Ehxin mathcal C_E : xx be the real Banach subspace of self-adjoint operators in (mathcal C_E, cdot_mathcal C_E). We show that in the case when (mathcal C_E, cdot_mathcal C_E) is a separable or perfect Banach symmetric ideal (mathcal C_E eq mathcal C_2) any skew-Hermitian operator H: mathcal C_Ehto mathcal C_Eh has the following form H(x)i(xa - ax) for same aain mathcal B(mathcal H) and for all xin mathcal C_Eh. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:mathcal C_Eh to mathcal C_Eh. Let (mathcal C_E, cdot_mathcal C_E) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is p_mathcal C_E 1 for any finite dimensional projection p inmathcal C_E with dim p(mathcal H)1, let mathcal C_E eq mathcal C_2, and let V: mathcal C_Eh to mathcal C_Eh be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on mathcal H such that V(x)uxu orV(x)-uxu for all x in mathcal C_Eh.

EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000227 ◽

2009 ◽

Vol 80 (1) ◽

pp. 83-90 ◽

Cited By ~ 1

Author(s):

SHUDONG LIU ◽

XIAOCHUN FANG

Keyword(s):

Hilbert Space ◽

Compact Operators ◽

Cuntz Algebra ◽

Infinite Dimensional ◽

Algebra Ii ◽

K Theory ◽

Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

Partial automorphisms of stable C-algebras and Hilbert C-bimodules

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010971 ◽

2005 ◽

Vol 79 (3) ◽

pp. 391-398

Author(s):

Kazunori Kodaka

Keyword(s):

Hilbert Space ◽

Compact Operators ◽

Equivalence Classes ◽

Infinite Dimensional ◽

C Algebras ◽

Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

On Products of Symmetries

10.4153/cjm-1966-090-5 ◽

1966 ◽

Vol 18 ◽

pp. 897-900 ◽

Cited By ~ 12

Author(s):

Peter A. Fillmore

Keyword(s):

Hilbert Space ◽

Von Neumann Algebra ◽

Unitary Operator ◽

Type Ii ◽

Central Projections ◽

Von Neumann ◽

Infinite Dimensional ◽

Principal Tool ◽

The Subject

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

On Self-Adjoint Factorization of Operators

10.4153/cjm-1969-156-6 ◽

1969 ◽

Vol 21 ◽

pp. 1421-1426 ◽

Cited By ~ 4

Author(s):

Heydar Radjavi

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Bounded Linear Operator ◽

Unitary Operator ◽

Normal Operator ◽

Complex Hilbert Space ◽

Bounded Linear ◽

Infinite Dimensional ◽

Normal Operators ◽

Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

On Compact Perturbations of Operators

10.4153/cjm-1974-024-3 ◽

1974 ◽

Vol 26 (1) ◽

pp. 247-250 ◽

Cited By ~ 1

Author(s):

Joel Anderson

Keyword(s):

Hilbert Space ◽

Compact Operator ◽

Unitary Operator ◽

General Theorem ◽

Infinite Dimensional ◽

Compact Perturbations ◽

Small Norm

Recently R. G. Douglas showed [4] that if V is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinite dimensional Hilbert space ), then V — K is unitarily equivalent to V ⊕ U (acting on ⊕ ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary).

Strong Extensions vs. Weak Extensions of C*-Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1978-025-4 ◽

1978 ◽

Vol 21 (2) ◽

pp. 143-147

Author(s):

S. J. Cho

Keyword(s):

Hilbert Space ◽

Partial Isometry ◽

Compact Operators ◽

Bounded Operators ◽

Infinite Dimensional ◽

C Algebras ◽

Image Position ◽

The Ideal

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such thatfor all a in A.

Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652005000400002 ◽

2005 ◽

Vol 77 (4) ◽

pp. 589-594 ◽

Cited By ~ 2

Author(s):

Paolo Piccione ◽

Daniel V. Tausk

Keyword(s):

Hilbert Space ◽

Hilbert Spaces ◽

Countable Family ◽

Spectral Theorem ◽

Dimensional Case ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Lagrangian Subspaces ◽

Finite Dimensional Case ◽

Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

On the statistical estimation of the logarithmic derivative of a measure in a Hilbert space

Georgian Mathematical Journal ◽

10.1515/gmj.2010.042 ◽

2010 ◽

Vol 17 (4) ◽

pp. 741-747

Author(s):

Elizbar Nadaraya ◽

Grigol Sokhadze

Keyword(s):

Hilbert Space ◽

Statistical Estimation ◽

Logarithmic Derivative ◽

Observation Data ◽

Independent Observation ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Approximating Sequence

Abstract We consider the problem of statistical estimation of the logarithmic derivative of a measure in an infinite-dimensional Hilbert space. It is shown that an approximating sequence of finite-dimensional estimates can be constructed for the unknown logarithmic derivative using independent observation data.

The Bishop-Phelps-Bollobàs Property for Compact Operators

10.4153/cjm-2016-036-6 ◽

2018 ◽

Vol 70 (1) ◽

pp. 53-57 ◽

Cited By ~ 5

Author(s):

Sheldon Dantas ◽

Domingo García ◽

Manuel Maestre ◽

Miguel Martín

Keyword(s):

Hilbert Space ◽

Topological Space ◽

Banach Spaces ◽

Positive Measure ◽

Compact Operators ◽

Function Spaces ◽

Sequence Spaces ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Nikodym Property

AbstractWe study the Bishop-Phelps-Bollobàs property (BPBp) for compact operators. We present some abstract techniques that allow us to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X and Y be Banach spaces. If (c0, Y) has the BPBp for compact operators, then so do (C0(L), Y) for every locally compactHausdorò topological space L and (X, Y) whenever X* is isometrically isomorphic to . If X* has the Radon-Nikodým property and (X), Y) has the BPBp for compact operators, then so does (L1(μ, X), Y) for every positive measure μ; as a consequence, (L1(μ, X), Y) has the BPBp for compact operators when X and Y are finite-dimensional or Y is a Hilbert space and X = c0 or X = Lp(v) for any positive measure v and 1 < p < ∞. For , if (X, (Y)) has the BPBp for compact operators, then so does (X, Lp(μ, Y)) for every positive measure μ such that L1(μ) is infinite-dimensional. If (X, Y) has the BPBp for compact operators, then so do (X, L∞(μ, Y)) for every σ-finite positive measure μ and (X, C(K, Y)) for every compact Hausdorff topological space K.

Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

10.4153/cjm-2011-092-x ◽

2012 ◽

Vol 64 (4) ◽

pp. 755-777 ◽

Cited By ~ 5

Author(s):

Lawrence G. Brown ◽

Hyun Ho Lee

Keyword(s):

Hilbert Space ◽

Compact Operators ◽

Homotopy Classification ◽

Multiplier Algebra ◽

Von Neumann ◽

C Algebra ◽

Infinite Dimensional ◽

Dimensional Hilbert Space

AbstractWe study projections in the corona algebra of C(X) ⊗ K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space and X = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in K0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.