linear operator

[′lin·ē·ər ′äp·ə‚rād·ər]

(mathematics)

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 x ∈ E, 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.

Mentioned in

References in periodicals archive

We have by Theorem 2.1 that [phi] is an invertible linear operator which preserves commuting pairs of matrices.

Linear operators preserving commuting pairs of matrices over semirings

Fortunately, the homotopy-series (12) contains an auxiliary parameter h, and besides we have great freedom to choose the auxiliary linear operator L, as illustrated by Liao (20).

An analytical scheme for multi-order fractional differential equations

where B is a homogeneous linear operator and A x B is the product of two linear operators.

Generalization of connection based on the concept of graded q-differential algebra/Gradueeritud q-diferentsiaalalgebrale tuginev seostuse uldistus

[16] A family of bounded linear operators R(t) [member of] B(X) for t [member of] J is called a resolvent operator for

Solvability of impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators

If T: X [right arrow] Y is a surjective bounded linear operator then T is sequentially continuous.

2-NSR lemma and compact operators on 2-normed space

For a space X, B(X) will denote the set of bounded linear operators on X.

The point spectra of generalized Hausdorff matrices

A densely defined linear operator T is called closed if its graph g(T) [subset] H [direct sum] H is closed ([1], section 46).

Approximation of bandlimited functions on a non-compact manifold by bandlimited functions on compact submanifolds

is a convolution linear operator on [F.sub.[theta]](N').

Evolution equation associated with the power of the Gross Laplacian

The matrix E[a] defines an invertible linear operator.

C++ Classes for Linking Optimization with Complex Simulations

(iv) When a closed densely defined linear operator A in a complex Banach space X generates a strongly continuous group [{T(t)}.sub.t[member of]R] of bounded linear operators (see, e.g., [3, 5]), i.e., the associated abstract Cauchy problem (ACP)

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

A bounded linear operator A is said to be quasi-Hermitian if its imaginary component

Stability for Linear Volterra Difference Equations in Banach Spaces