closed operator

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closed operator

[¦klōzd ′äp·ə‚rād·ər]
(mathematics)
A linear transformation ƒ whose domain A is contained in a normed vector space X satisfying the condition that if lim xn = x for a sequence xn in A, and lim ƒ(xn) = y, then x is in A and ƒ(x) = y.
References in periodicals archive ?
1) takes values in a Banach space X (state space), the unbounded operator (A,D(A)) generates a [C.
Thus the bounded integral operator C (t) (D (t)) with |t| < 1 gives the right (left) inverse of the unbounded operator A ([t.
If S: D(S) [subset] K [right arrow] L is another unbounded operator, then ST: D(ST) [subset] H [right arrow] L is a new operator for which D(ST) = {x [member of] D(T): Tx [member of] D(S)} and (ST)x = S(Tx) for all x [member of] D(ST).
In [19] there is an example of a linear but unbounded operator T which is hypercyclic but neither [T.
In what follows, we assume that {A(t), t [greater than or equal to] 0} is a family of closed densely defined linear unbounded operators on the Banach space E and with domain D(A(t)) independent of t.
Triggiani: Extensions of rank conditions for controllability and observability to Banach space and unbounded operators, SIAM J.
of Berlin) presents iterative splitting methods for decomposing evolution equations and decoupling differential equations with unbounded operators.
She covers normed spaces and operators, Frechet spaces and Banach theorems, duality, weak topologies, distributions, the Fourier transform and Sobolev spaces, Banach algebras, and unbounded operators in a Hilbert space.
The case with an unbounded non-selfadjoint operator as a main operator in a corresponding time invariant system has not been investigated sufficiently because of a number of essential difficulties usually related to unbounded operators (especially non-selfadjoint ones).
Three appendices develop the theory of ultrapowers, discuss the basic theory of unbounded operators, and provide an alternative approach from that of the main text to the existence of the trace in the basic construction.
The last result in this paper is a generalization of Theorem 3 (see below) to unbounded operators.