Criteria for completeness of the system of root vectors of a dissipative operator

1963 ◽  
pp. 221-229 ◽  
Author(s):  
M. G. Kreĭn
Keyword(s):  
Dissipative Operator

scholarly journals Uniqueness criterion for solution of abstract nonlocal Cauchy problem

1993 ◽  
Vol 6 (1) ◽  
pp. 49-54 ◽  
Author(s):  
L. Byszewski
Keyword(s):  
Cauchy Problem ◽  
Nonlocal Condition ◽  
Nonlocal Problem ◽  
Dissipative Operator ◽  
Uniqueness Criterion ◽  
Nonlocal Cauchy Problem

The aim of the paper is to prove an uniqueness criterion for a solution of an abstract nonlocal Cauchy problem. A dissipative operator in the nonlocal problem and an arbitrary functional in the nonlocal condition are considered. The paper is a continuation of papers [1]-[3] and generalizes some results from [4].

Linear Combinations of the Volterra Dissipative Operator and Its Adjoint Operator

2013 ◽  
Vol 65 (5) ◽  
pp. 780-786
Author(s):  
G. M. Gubreev ◽  
E. I. Olefir ◽  
A. A. Tarasenko
Keyword(s):  
Adjoint Operator ◽  
Dissipative Operator ◽  
Linear Combinations

On inequalities for powers of linear operators and for quadratic forms

1981 ◽  
Vol 89 (1-2) ◽  
pp. 25-50 ◽  
Author(s):  
Vũ Qúôc Phóng
Keyword(s):  
Quadratic Forms ◽  
Linear Operators ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Domain ◽  
The Best Possible Constant ◽  
Bilinear Functional ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLetHbe a Hilbert space in which a symmetric operatorSwith a dense domainDsis given and letShave a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov typeand a method for calculating the best possible constantsCn,m(S).Moreover, let φ be a symmetric bilinear functional with a dense domainDφsuch thatDs⊂Dφand φ(f, g) = (Sf, g) for allf∈Ds,g∈Dφ. A necessary and sufficient condition for validity of the inequalityas well as a method for calculating the best possible constantKare obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the formThe paper is concluded by establishing the best possible constants in the inequalitieswhereTis an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

p Dissipative Operator

1999 ◽  
Vol 201 (3) ◽  
pp. 519-548 ◽  
Author(s):  
Lixin Tian ◽  
Zengrong Liu
Keyword(s):  
Dissipative Operator

Dissipative operators with finite dimensional damping

1982 ◽  
Vol 91 (3-4) ◽  
pp. 243-263 ◽  
Author(s):  
Harald Röh
Keyword(s):  
Separable Hilbert Space ◽  
Dissipative Operator ◽  
Finite Codimension ◽  
Spectral Mapping Theorem ◽  
Damped Wave Equation ◽  
Spectral Mapping ◽  
Dissipative Operators ◽  
Finite Dimensional ◽  
Maximal Dissipative Operator ◽  
Cayley Transformation

SynopsisLetG: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ andT(t) be theCosemigroup generated byG. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {xε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions forGare satisfied. Proofs are given in terms of the Cayley transformationT= (G+I)(G−I)−1ofG. The results are applied to the damped wave equationutt+ γutx+uxxxx+ ßuxx= 0, 0 ≦t< ∞ 0 <x< 1, β, γ ≧ 0, with boundary conditionsu(0,t) =ux(0,t) =uxx(1,t) =uxxx(1,t) = 0.

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