By Lajos Molnár (auth.)
Over the previous numerous a long time, the territory of preserver difficulties has been consistently enlarging in the body of linear research. the purpose of this paintings is to offer a kind of cross-section of the fashionable idea of preservers on endless dimensional areas (operator areas and serve as areas) in the course of the author's corresponding effects. designated emphasis is wear preserver difficulties touching on a few buildings of Hilbert area operators which look in quantum mechanics. furthermore, neighborhood automorphisms and native isometries of operator algebras and serve as algebras are mentioned in details.
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Sample text
The operator P ∈ B(X) is called an idempotent if P 2 = P . Let H be a Hilbert space. , . If x, y ∈ H, then x ⊗ y stands for the operator defined by (x ⊗ y)z = z, y x (z ∈ H). If A ∈ B(H), then its adjoint (in the Hilbert space sense) is denoted by A∗ . Fixing an arbitrary complete orthonormal system in H and considering the corresponding matrix representation of operators, one can define the transpose Atr of an arbitrary operator A ∈ B(H) in the obvious way. The operator P ∈ B(H) is called a projection if it is a self-adjoint idempotent.
The relation ker Q ⊂ ker P can be proved in a similar manner. Using the above characterization and the preserving property of φ, we obtain that φ preserves the relation ≤ between regular idempotents. Now, if R is a finite rank idempotent, then R can be written in the form R = Q − P with some regular idempotents P ≤ Q. Since φ(P ) ≤ φ(Q), it follows that φ(R) = φ(Q) − φ(P ) is also an idempotent. We prove that φ(R) is of finite rank. Choosing a regular idempotent P with R ≤ P , it follows that P − R is a regular idempotent and hence φ(P ) − φ(R) is also an idempotent.
Any subalgebra of B(X) which contains F (X) is called a standard operator algebra on X. If A ∈ B(X), then ker A denotes the kernel of A while rng A stands for the range of A. The spectrum of A is denoted by σ(A). The operator P ∈ B(X) is called an idempotent if P 2 = P . Let H be a Hilbert space. , . If x, y ∈ H, then x ⊗ y stands for the operator defined by (x ⊗ y)z = z, y x (z ∈ H). If A ∈ B(H), then its adjoint (in the Hilbert space sense) is denoted by A∗ . Fixing an arbitrary complete orthonormal system in H and considering the corresponding matrix representation of operators, one can define the transpose Atr of an arbitrary operator A ∈ B(H) in the obvious way.