# Linear Operator

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Related to Linear Operator: Linear differential operator

## linear operator

[′lin·ē·ər ′äp·ə‚rād·ər]
(mathematics)

## Linear Operator

a generalization of the concept of linear transformation to vector spaces. F is called a linear operator on a vector space E if it is a function on E with values in some vector space E1 and has the linearity property

F(αx + βy) = αF(x) + βF(y)

where x and y are elements of E, and α and β are numbers. If the E and E1 are normed spaces and ǀǀF(x)ǀǀ / ǀǀxǀǀ is uniformly bounded for all xE, then the linear operator F is said to be bounded and is called its norm.

The most important linear operators on function spaces are the differential linear operators and the integral linear operators The Laplace operator is an example of a linear operator on a space of functions of many variables. The theory of linear operators finds numerous applications in various problems of mathematical physics and applied mathematics.

References in periodicals archive ?
In this paper we study problem (1) for the case where the bounded linear operator A is singular but g-Drazin invertible.
If the auxiliary linear operator, the initial guesses, the auxiliary parameters h1, h2 and the auxiliary functions [H.
Dziok and Srivastava , using Wright's generalized hypergeometric function , Dziok and Raina  defined another linear operator given by
3] if A is a closed linear operator with domain D(A) defined on a Banach space E and [alpha] > 0, the we say that A is the generator of an [alpha]-resolvent family if there exists [omega] [greater than or equal to] 0 and a strongly continuous function [S.
It is also observed that a better choice of auxiliary linear operator increases the region of convergence.
Obviously, when q = 0, because of the property L(0) = 0 of any linear operator L, Eqs.
If F [right arrow] F is a homogeneous linear operator, we can extend it to the linear operator [L.
alpha]], for 0 < [alpha] [less than or equal to] 1, as a closed linear operator on its domain D([A.
As usual, 23(H, X) is the Banach space of all bounded linear operators from H into X, while [?
A linear operator T: X [right arrow] Y is called a compact operator if it maps every locally bounded sequence {[x.
Given a linear operator, T, defined on a dense linear subspace, D(T), of a separable Hilbert space, H, consider the domain of its adjoint:

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