Continuous Linear Maps

2014 ◽  
pp. 115-138
Author(s):  
Joseph Muscat
Keyword(s):  
Continuous Linear ◽  
Linear Maps

scholarly journals On the surjectivity of linear maps on locally convex spaces

1982 ◽  
Vol 32 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Basic Notion ◽  
Linear Map ◽  
Linear Maps ◽  
Locally Convex ◽  
Do So ◽  
Convex Spaces

AbstractThe aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

A representation theory of continuous linear maps

1964 ◽  
Vol 155 (4) ◽  
pp. 270-291 ◽  
Author(s):  
Khyson Swong
Keyword(s):  
Representation Theory ◽  
Continuous Linear ◽  
Linear Maps

scholarly journals Pitt's theorem for operators between general Lorentz sequence spaces

2002 ◽  
Vol 90 (1) ◽  
pp. 101 ◽  
Author(s):  
J. A. López Molina
Keyword(s):  
Sequence Spaces ◽  
Continuous Linear ◽  
Linear Maps ◽  
Pitt’S Theorem

We characterize the pairs of general Lorentz sequence spaces $\ell^{u,v}(\nu),$ $\ell^{p,q}(\mu), 0 < u, v, p, q < \infty$ such that all continuous linear maps from the first space into the second one are compact.

scholarly journals Closed linear maps from a barrelled normed space into itself need not be continuous

1998 ◽  
Vol 57 (2) ◽  
pp. 177-179 ◽  
Author(s):  
José Bonet
Keyword(s):  
Normed Space ◽  
Continuous Linear ◽  
Closed Graph ◽  
Linear Map ◽  
Linear Maps ◽  
Barrelled Spaces

Examples of normed barrelled spacesEor quasicomplete barrelled spacesEare given such that there is a non-continuous linear map from the spaceEinto itself with closed graph.

Topological decompositions of the duals of locally convex operator spaces

1983 ◽  
Vol 93 (2) ◽  
pp. 307-314 ◽  
Author(s):  
D. J. Fleming ◽  
D. M. Giarrusso
Keyword(s):  
Uniform Convergence ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Natural Topology ◽  
Usual Norm ◽  
Linear Maps ◽  
Locally Convex

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

scholarly journals Linear Maps Which are Anti-derivable at Zero

2020 ◽  
Vol 43 (6) ◽  
pp. 4315-4334
Author(s):  
Doha Adel Abulhamil ◽  
Fatmah B. Jamjoom ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Linear Operators ◽  
Continuous Linear ◽  
Bounded Linear ◽  
Linear Maps ◽  
Continuous Linear Operators

Abstract Let $$T:A\rightarrow X$$ T : A → X be a bounded linear operator, where A is a $$\hbox {C}^*$$ C ∗ -algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., $$ab =0$$ a b = 0 in A implies $$T(b) a + b T(a)=0$$ T ( b ) a + b T ( a ) = 0 ); (b) There exist an anti-derivation $$d:A\rightarrow X^{**}$$ d : A → X ∗ ∗ and an element $$\xi \in X^{**}$$ ξ ∈ X ∗ ∗ satisfying $$\xi a = a \xi ,$$ ξ a = a ξ , $$\xi [a,b]=0,$$ ξ [ a , b ] = 0 , $$T(a b) = b T(a) + T(b) a - b \xi a,$$ T ( a b ) = b T ( a ) + T ( b ) a - b ξ a , and $$T(a) = d(a) + \xi a,$$ T ( a ) = d ( a ) + ξ a , for all $$a,b\in A$$ a , b ∈ A . We also prove a similar equivalence when X is replaced with $$A^{**}$$ A ∗ ∗ . This provides a complete characterization of those bounded linear maps from A into X or into $$A^{**}$$ A ∗ ∗ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are $$^*$$ ∗ -anti-derivable at zero.

A Uniqueness Theorem on the Degenerate Cauchy Problem(1)

1975 ◽  
Vol 18 (3) ◽  
pp. 417-421 ◽  
Author(s):  
Chung-Lie Wang
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Uniqueness Theorem ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Continuous Linear ◽  
Linear Maps ◽  
Image Position ◽  
Degenerate Cauchy Problem ◽  
Densely Defined

In [4] Carroll and the author have treated the following problem(1)where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).

The idempotent generated subsemigroup of the semigroup of continuous endomorphisms of a separable Hilbert space

1983 ◽  
Vol 94 (3-4) ◽  
pp. 351-360 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Null Space ◽  
Continuous Linear ◽  
Linear Maps ◽  
Image Position

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {x∊H: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, whereand ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.

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