Space Of Continuous Linear Operators Research Articles

ON BOUNDED WEAK AND PSEUDO-SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS HAVING TRICHOTOMY WITH AND WITHOUT DELAY IN BANACH SPACES

In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation [Formula: see text] where A:ℝ → L(E) is strongly measurable and Bochner integrable function on every finite subinterval of ℝ with f:ℝ × E → E is only assumed to be weakly weakly sequentially continuous or Pettis-integrable and the linear equation [Formula: see text] has a trichotomy with constants α ≥ 1 and σ > 0, if x is absolutely continuous function and for each x* ∈ E* there exists a negligible set ℵx* such that for each t ∉ ℵx*, then we have [Formula: see text] We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations [Formula: see text] Let T, r, d > 0, Br = {x > E: ‖x‖ ≤ r} and CE([-d,0]) be the Banach space of continuous functions from [-d,0] into E. Finally we prove an existence result for the differential equation with delay [Formula: see text] where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function, [Formula: see text] is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].

$\delta $-function of an operator: A white noise approach

Let (E) ⊂ (L 2 ) ⊂ (E)* be the canonical framework of white noise analysis over the Gel'fand triple S(R) ⊂ L 2 (R) ⊂ S*(R) and L ≡ L[(E),(E)*] be the space of continuous linear operators from (E) to (E)*. Let Q be a self-adjoint operator in (L 2 ) with spectral representation Q = f R λ P Q (dλ). In this paper, it is proved that under appropriate conditions upon Q, there exists a unique linear mapping Z: S*(R) → L such that Z(f) = ∫ R f(λ)P Q (dλ) for each f ∈ S(R). The mapping is then naturally used to define δ(Q) as Z(δ), where δ is the Dirac δ-function. Finally, properties of the mapping Z are investigated and several results are obtained.

ORLICZ - PETTIS THEOREMS FOR MULTIPLIER CONVERGENT OPERATOR VALUED SERIES

Let X, Y be locally convex spaces and L(X, Y ) the space of continuous linear operators from X into Y . We consider 2 types of multiplier convergent theorems for a series ∑ Tₕ in L(X, Y ). First, if λ is a scalar sequence space, we say that the series ∑ Tₕ is λ multiplier P convergent for a locally convex topology τ on L(X, Y ) if the series ∑ tₕTₕ is τ convergent for every t = {tₕ} ∈ λ. We establish conditions on λ which guarantee that a λ multiplier convergent series in the weak or strong operator topology is λ multiplier convergent in the topology of uniform convergence on the bounded subsets of X. Second, we consider vector valued multipliers. If E is a sequence space of X valued sequences, the series ∑ Tₕ is E multiplier convergent in a locally convex topology η on Y if the series ∑ Tₕxₕ is η convergent for every x = {xₕ} ∈ E. We consider a gliding hump property on E which guarantees that a series ∑ Tₕ which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y .

The Composition of Operator-Valued Measurable Functions is Measurable

Given separable Frechet spaces, E, F, and G, let $\mathcal {L}(E,F),\mathcal {L}(F,G)$, and $\mathcal {L}(E,G)$ denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if $(X,\mathcal {F})$ is any measurable space and both $A:X \to \mathcal {L}(E,F)$ and $B:X \to \mathcal {L}(F,G)$ are Borelian, then the operator composition $BA:X \to \mathcal {L}(E,G)$ is also Borelian. Further, we will give several consequences of this result.

The composition of operator-valued measurable functions is measurable

Given separable Frechet spaces, E, F, and G, let L ( E , F ) , L ( F , G ) \mathcal {L}(E,F),\mathcal {L}(F,G) , and L ( E , G ) \mathcal {L}(E,G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if ( X , F ) (X,\mathcal {F}) is any measurable space and both A : X → L ( E , F ) A:X \to \mathcal {L}(E,F) and B : X → L ( F , G ) B:X \to \mathcal {L}(F,G) are Borelian, then the operator composition B A : X → L ( E , G ) BA:X \to \mathcal {L}(E,G) is also Borelian. Further, we will give several consequences of this result.

Isometric properties of the subspace of compact operators in a space of continuous linear operators

Isometric properties of the subspace of compact operators in a space of continuous linear operators

Remarks on completeness in spaces of linear operators

Whereas a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is not so in the converse situation and is the problem discussed here. The barrelledness of X in its Mackey topology plays an important role: if L (X) is quasicomplete, then X is barrelled for its Mackey topology. Consequently, for Mackey spaces X is turns out that L (X) is quasicomplete if and only if X is quasicomplete and barrelled: this is false if sequential completeness is substituted for quasicompleteness. Furthermore, there exist non-barrelled spaces X for which X and L (X) are quasicomplete (sequentially complete). Hence, although barrelledness is a sufficient condition for completeness of L (X) in various senses, it is certainly not necessary.

The Completion of the Maximal Op*-Algebra on a Frechet Domain

This paper investigates the completion of the maximal Op*-algebra L + ( D ) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D + denote the strong dual of D , equipped with the complex conjugate linear structure. It is shown that the completion of L + ( D ) (endowed with the uniform topology) is the space of continuous linear operators ??? ( D , <7>D+) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterization of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in L + ( D ).

On whether or not [formula omitted]( E, F) = [formula omitted]r( E, F) for some classical Banach lattices E and F

For Banach lattices E and F, L ( E,F) is the space of all continuous linear operators E→ F, L r( E,F) is the vector space of all regular continuous linear operators E→ F which is endowed with the r-norm. This paper concerns the problems: (1) is every continuous linear operator E→F regular? (2) if the answer to (1) is “yes”, there is a further problem: is its operator norm in L ( E,F) equal to its r-norm in L r( E,F)? A series of conclusions is obtained for cases in which each of E and F is one of Banach lattices l p (1≤ p < ∞), l ∞, c 0, c, C[0,1] and C( X).

M-Structure in the Banach Algebra of Operators on C 0 (Ω)

The M-ideals in B(CO(Q)), the space of continuous linear operators on CO(Q), are determined where Q is a locally compact Hausdorff countably paracompact space. A one-to-one correspondence between M-ideals in B(C0( Q)), open subsets of the Stone-Cech compactification of Q, and lower semicontinuous Hermitian projections in B(C0(Q))** is established.

Analytic perturbation of semi-fredholm operators

Let X and Y be locally convex Hausdorff spaces and L(X, Y) the space of continuous linear operators from X to Y. If TeL(X, Y), we denote by N(T) its kernel or null space and by R(T) its range, q~-(X, Y) is defined as the set of all operators TeL(X, Y) which are open, have a closed range and finite nullity nul(T) := dimN(T). Mappings of this type are called lower semi-Fredholm or ~ operators, q)+(X, Y) is defined as the set of those operators TeL(X, Y) which are open and have a closed range of finite codimension, i.e. they have deficiencies clef(T) := codim R(T)< oo. Operators of this kind are called upper semi-Fredholm or q~+-operators. If T is a lower or upper semi-Fredholm operator, then the index of T is defined as ind(T):=nul(T)-def(T). An operator which belongs to 9 (X, Y):= ~-(X, Y)c~+(X, Y) is said to be a Fredholm or q~-operator. We denote by ~Z(X, Y) or ~r(X, Y) the subset of those operators in ~-(X, Y) or ~+(X, Y) respectively which have projectable kernels and ranges; these operators are called ~,. or q~r-operators. In this present paper G is a region in C N. We consider H(G, L, (X, Y)), the family of holomorphic maps from G to L(X, Y), where L(X, Y) is provided with the topology of pointwise convergence. Let H(G, q~• Y)) be the subset of those functions in H(G, L(X, Y)) which have values in ~+(X, Y) or ~-(X, Y) respectively. We consider the set of sinoularities

Tensor products and almost periodicity

Let E and F be locally convex spaces and G their completed e-tensor product. It is shown that if S and T are weakly almost periodic equicontinuous semigroups of operators on E and F respectively, then, under mild restrictions on E or F, SoT is a weakly almost periodic equicontinuous semigroup of operators on G, and the almost periodic and flight vector subspaces of G are related in a natural way to the corresponding subspaces of E and Fvia the e-tensor product. Furthermore, if Eand Fboth decompose into a direct sum of these subspaces then so does G. 1. Weakly almost periodic semigroups. Let E be a locally convex (Hausdorff) linear topological space with topological dual E', and let L(E) denote the space of continuous linear operators on E. A semigroup of operators on E is a subset S of L(E) containing the identity operator and closed under composition. A vector x E E is said to be weakly (strongly) almost periodic under S if its orbit Sx= {ux: u E S} is relatively compact in the weak (strong) topology of E. The set of all weakly (strongly) almost periodic vectors in E shall be denoted by W(E, S) (A(E, S)). Occasionally we shall suppress the symbols E or S from the notation if they are understood from context. If W(E, S)=E (A(E, S)=E) we say that S is weakly (strongly) almost periodic. It is easily seen that multiplication in a semigroup of operators S on a locally convex space E is separately continuous with respect to the weak or strong operator topologies on L(E), that is to say S is a topological semigroup. Moreover if S is equicontinuous then multiplication is actually jointly continuous in the strong operator topology. The following lemma is at the heart of the theory of weak almost periodicity. A proof can be found in [1]. LEMMA 1.1. Let S be a weakly almost periodic equicontinuous semigroup of operators on a locally convex space E, and let 3 denote the closure Received by the editors March 26, 1973 and, in revised form June 21, 1973. AMS (MOS) subject classiflcations (1970). Primary 47D05, 46M05; Secondary 43A60, 22A15.

An elementary characterization of absolutely summing operators

(l 1 IX], =1) is also a Banach space. Denote by • (X; ~ ) the space of continuous linear operators from X into ~ . We say that u ~ ~ ( X ; N) is absolutely summin 9 whenever u "lifts" to a continuous linear operator from la(X) into 11[~ ], i.e., (u(xn)) ~ I1 [N] for all (x,) e tl(X), the continuity of the resulting operation follows from [73, Theorem 4. Let (f2, S, #) be a finite, positive, non-atomic measure space. Denote by ~ ( # , X ) the class of all strongly measurable Pettis #-integrable functions f : t2-*X; norm U;;~(#, X) by

The space of continuous linear operators as a completion of $E' \otimes F$

The space of continuous linear operators as a completion of $E' \otimes F$

Fredholm operators: A counter-example

In 1960 Pietsch [I] gave an example of a locally convex linear topological space for which the set of Fredholm operators is not open. It is the purpose of this note to show that the Fredholm class is not open for any weak locally convex space. A Hausdorff locally convex space is called a weak locally convex space if every null-neighborhood contains a closed subspace of finite codimension. A continuous linear operator T mapping a linear topological space X into itself is called a Fredholm operator if it enjoys the following properties: (1) The null space N(T) is finite dimensional, (2) the range space R(T) is finite codimensional, (3) T is relatively open, and (4) R(T) is closed. The class of Fredholm operators on X is denoted by X(X). Let X be a Hausdorff locally convex space and L(X) the space of continuous linear operators on X. L(X) is topologized with the topology of uniform convergence on bounded subsets of X.

Integral representation of multiplicative, involution preserving operators in ℒ(𝒞(𝒮),ℰ)

1. Introduction. Throughout, S will be a compact Hausdorff space and E will be a Banach space which is the dual of another Banach space F. C(S) will denote the space of complex-valued continuous functions on S topologized with the topology of uniform convergence. ?(C(S), E) will denote the space of continuous linear operators from C(S) to E. A theorem of Gil de Lamadrid [5, p. 103] identifies ?(C(S), E) with a space of E-valued measures, the correspondence between operator and measure being given by integration. A closely related result was given earlier by Bartle, Dunford, and Schwartz [2 ]. Now if E is a Banach algebra with involution [7, p. 178], it makes sense to consider operators which are not only continuous and linear but which also preserve multiplication and involution. A natural question arises: How are these additional properties reflected in the representing measure? We answer this question under additional restrictions on E. We also give several examples of spaces E satisfying the hypotheses of our theorems. One can use the results of this paper to prove the Spectral Theorem for bounded operators; but the proof follows standard lines and will not be included (see [6, p. 99]). We conclude this introduction with a precise description of Gil de Lamadrid's Theorem. Our description differs from Gil de Lamadrid's, but it is not difficult to verify that they are equivalent. We consider the class N(S, E) of set functions m from 6((S), the Borel class of S, to E which are countably additive and regular with respect to the weak topology a(E, F) on E induced by F. To say that m: (3(S) -E is regular with respect to the topology o(E, F) means that, for every BE@(S) and every a(E, F)-neighborhood N of 0, there exists a compact set K and an open set U such that KCBC U and, if A C U-K, then m(A) EN. Defining addition and scalar-multiplication in N(S, E) in the usual set-wise fashion; i.e., (m1+m2)(B)=m1(B) +m2(B) and (am,)(B) ==am(B), N(S, E) is a vector space. The following formula defines a norm on N(S, E) making it a Banach space: