Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

1985 ◽  
Vol 52 (1-2) ◽  
pp. 15-27 ◽  
Author(s):  
S. J. Dilworth
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Compact Operators ◽  
Reflexive Banach Spaces

scholarly journals Derivations on commutative operator algebras

1985 ◽  
Vol 32 (3) ◽  
pp. 415-418
Author(s):  
Mark Spivack
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Bounded Operator ◽  
Alternative Proof ◽  
Reflexive Banach Spaces ◽  
Von Neumann ◽  
Simple Alternative

It is well-known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator. In this note we present a simple alternative proof, which generalizes the result further within Hilbert space, and to reflexive Banach spaces.

scholarly journals Spaces of compact operators which areM-ideals inL(X,Y)

1992 ◽  
Vol 15 (3) ◽  
pp. 617-619
Author(s):  
Chong-Man Cho
Keyword(s):  
Banach Spaces ◽  
Dual Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Reflexive Banach Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Second Dual

SupposeXandYare reflexive Banach spaces. IfK(X,Y), the space of all compact linear operaters fromXtoYis anM-ideal inL(X,Y), the space of all bounded linear operators fromXtoY, then the second dual spaceK(X,Y)**ofK(X,Y)is isometrically isomorphic toL(X,Y).

scholarly journals The Bishop-Phelps-Bollobàs Property for Compact Operators

2018 ◽  
Vol 70 (1) ◽  
pp. 53-57 ◽  
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.

Some surjectivity results for operators of generalized monotone type via a topological degree

2018 ◽  
Vol 34 (3) ◽  
pp. 333-340
Author(s):  
SUK-JOON HONG ◽  
◽  
IN-SOOK KIM ◽  
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Topological Degree ◽  
Degree Theory ◽  
Monotone Operators ◽  
Reflexive Banach Spaces ◽  
Space Case ◽  
Monotone Type

We introduce a topological degree for a class of operators of generalized monotone type in reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we give some surjectivity results for operators of generalized monotone type in reflexive Banach spaces. In the Hilbert space case, this reduces to the celebrated Browder-Minty theorem for monotone operators.

scholarly journals CO-REPRESENTATIONS OF HOPF–VON NEUMANN ALGEBRAS ON OPERATOR SPACES OTHER THAN COLUMN HILBERT SPACE

2010 ◽  
Vol 82 (2) ◽  
pp. 205-210 ◽  
Author(s):  
VOLKER RUNDE
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Spaces ◽  
Reflexive Banach Spaces ◽  
Von Neumann ◽  
Reflexive Operator

AbstractRecently, Daws introduced a notion of co-representation of abelian Hopf–von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf–von Neumann algebras. The key is our observation that, for a von Neumann algebra 𝔐 and a reflexive operator space E, the normal spatial tensor product $\M \btensor \CB (E)$ is a Banach algebra if and only if 𝔐 is subhomogeneous or E is completely isomorphic to column Hilbert space.

scholarly journals Operators constructed by means of basic sequences and nuclear matrices

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ahmed Morsy ◽  
Nashat Faried ◽  
Samy A. Harisa ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Schauder Basis ◽  
Countable Number ◽  
Nuclear Operator ◽  
Approximation Numbers ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Nuclear Operators

AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.

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