scholarly journals Identifications for General Degenerate Problems of Hyperbolic Type in Hilbert Spaces

2018 ◽  
Vol 64 (1) ◽  
pp. 194-210
Author(s):  
A Favini ◽  
G Marinoschi ◽  
H Tanabe ◽  
Ya Yakubov
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Hyperbolic Type ◽  
Closed Linear Operator ◽  
Abstract Problem ◽  
Strict Solution ◽  
Additional Information ◽  
Second Order Equations

In a Hilbert space X, we consider the abstract problem M∗ddt(My(t))=Ly(t)+f(t)z,0≤t≤τ,My(0)=My0, where L is a closed linear operator in X and M∈L(X) is not necessarily invertible, z∈X. Given the additional information Φ[My(t)]=g(t) wuth Φ∈X∗, g∈C1([0,τ];C). We are concerned with the determination of the conditions under which we can identify f∈C([0,τ];C) such that y be a strict solution to the abstract problem, i.e., My∈C1([0,τ];X), Ly∈C([0,τ];X). A similar problem is considered for general second order equations in time. Various examples of these general problems are given.

Existence of Solutions of Abstract Differential Equations in a Local Space

1973 ◽  
Vol 16 (2) ◽  
pp. 239-244
Author(s):  
M. A. Malik
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Differential Equations ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Existence Of Solutions ◽  
Closed Linear Operator ◽  
Abstract Differential Equations ◽  
Local Space

Let H be a Hilbert space; ( , ) and | | represent the scalar product and the norm respectively in H. Let A be a closed linear operator with domain DA dense in H and A* be its adjoint with domain DA*. DA and DA*are also Hilbert spaces under their respective graph scalar product. R(λ; A*) denotes the resolvent of A*; complex plane. We write L = D — A, L* = D — A*; D = (l/i)(d/dt).

Uniqueness of Generalized Solutions of Abstract Differential Equations

1975 ◽  
Vol 18 (3) ◽  
pp. 379-382
Author(s):  
M. A. Malik
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Differential Equations ◽  
Open Subset ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Complex Hilbert Space ◽  
Generalized Solutions ◽  
Closed Linear Operator ◽  
Abstract Differential Equations

Let Ω be an open subset of R and H be a complex Hilbert space; (,) represents scalar product in H.Let also A be a closed linear operator with domain DA dense in H and A* with domain D*A be its adjoint. Under graph scalar product DA and D*A are also Hilbert spaces.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

scholarly journals Pseudospectra in a non-Archimedean Banach space and essential pseudospectra in Eω

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3961-3976
Author(s):  
Aymen Ammar ◽  
Ameni Bouchekoua ◽  
Aref Jeribi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Completely Continuous ◽  
Completely Continuous Operator

In this work, we introduce and study the pseudospectra and the essential pseudospectra of linear operators in a non-Archimedean Banach space and in the non-Archimedean Hilbert space E?, respectively. In particular, we characterize these pseudospectra. Furthermore, inspired by T. Diagana and F. Ramaroson [12], we establish a relationship between the essential pseudospectrum of a closed linear operator and the essential pseudospectrum of this closed linear operator perturbed by completely continuous operator in the non-Archimedean Hilbert space E?.

scholarly journals Additive perturbations and multiplicative perturbations for the core inverse of bounded linear operator in Hilbert space

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6131-6144
Author(s):  
Fapeng Du ◽  
Zuhair Nashed
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Bounded Linear ◽  
The Core ◽  
Core Inverse

In this paper, we present some characteristics and expressions of the core inverse A# of bounded linear operator A in Hilbert spaces. Additive perturbations of core inverse are investigated under the condition R( ?)?N(A#) = {0} and an upper bound of ||?#-A#|| is obtained. We also discuss the multiplicative perturbations. The expressions of core inverse of perturbed operator T = EAF and the upper bounds of ||T#-A#|| are obtained too.

scholarly journals AN OPTIMAL LINEAR FILTER FOR ESTIMATION OF RANDOM FUNCTIONS IN HILBERT SPACE

2021 ◽  
pp. 1-28
Author(s):  
PHIL HOWLETT ◽  
ANATOLI TOROKHTI
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Random Vector ◽  
Optimal Filter ◽  
Linear Filter ◽  
Closed Linear Operator ◽  
Random Functions ◽  
Best Estimate ◽  
Covariance Operators ◽  
Optimal Linear

Abstract Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.

scholarly journals An optimal linear filter for estimation of random functions in Hilbert space

2021 ◽  
Vol 62 ◽  
pp. 274-301
Author(s):  
Phil George Howlett ◽  
Anatoli Torokhti
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Random Vector ◽  
Optimal Filter ◽  
Linear Filter ◽  
Closed Linear Operator ◽  
Random Functions ◽  
Best Estimate ◽  
Covariance Operators ◽  
Optimal Linear

Let \(\boldsymbol{f}\) be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space \(H\), and let \(\boldsymbol{g}\) be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space \(K\). We seek an optimal filter in the form of a closed linear operator \(X\) acting on the observable realizations of a proximate vector \(\boldsymbol{f}_{\epsilon} \approx \boldsymbol{f}\) that provides the best estimate \(\widehat{\boldsymbol{g}}_{\epsilon} = X\! \boldsymbol{f}_{\epsilon}\) of the vector \(\boldsymbol{g}\). We assume the required covariance operators are known. The results are illustrated with a typical example.   doi:10.1017/S1446181120000188

scholarly journals A simple proof of the closed graph theorem

2015 ◽  
Vol 4 (1) ◽  
pp. 1
Author(s):  
Alexander G. Ramm
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Simple Proof ◽  
Uniform Boundedness ◽  
Classical Theorem ◽  
Closed Linear Operator ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Uniform Boundedness Principle

<p>Assume that <em>A</em> is a closed linear operator defined on all of a Hilbert space <em>H</em>. Then, <em>A</em> is bounded. This classical theorem is proved on the basis of uniform boundedness principle. The proof is easily extended to Banach spaces.</p>

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