Let be an unbounded operator on a Hilbert space . An unbounded operator is its adjoint if
An adjoint does not need to exist in general.
An unbounded operator is symmetric if and for all (one also writes ).
The domain of is the set of all vectors such that the linear functional is bounded on .
The graph satisfies where denotes the orthogonal complement and denotes the transposition of the direct summands changing the sign of one of the factors, i.e. . An unbounded operator is closed if is closed subspace of . An operator is a closure of an operator if is a closure of operator . It is said that is an extension of and one writes if . The closure of an unbounded operator does not need to exist.
For any unbounded operator with a dense , if the adjoint operator exists, then is closed, and if exists then it coincides with a closure of .
An unbounded operator on a Hilbert space is self-adjoint if
An (unbounded) operator is essentially self-adjoint if it is symmetric and its spectrum (as a subspace of the complex plane) is contained in the real line. Alternatively, it is symmetric if its closure is self-adjoint.
A Hermitean (or hermitian) operator is a bounded symmetric operator (which is necessarily self-adjoint), although some authors use the term for any self-adjoint operator.
For a bounded operator between Hilbert spaces, define the Hermitean conjugate operator by , for all , . Distinguish it from the concept of the transposed operator? between the dual spaces.
In an arbitrary -algebra, a self-adjoint or hermitian element is any element such that .
Original articles:
See also:
A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
A. B. Antonevič, Ja. B. Radyno, Funkcional’nij analiz i integral’nye uravnenija, Minsk 1984
S. Kurepa, Funkcionalna analiza, elementi teorije operatora, Školska knjiga, Zagreb 1981.
Reed, M.; Simon, B.: Methods of modern mathematical physics. Volume 1, Functional Analysis
Walter Rudin, Functional analysis
Konrad Schmuedgen, Unbounded self-adjoint operators on Hilbert space, Springer GTM 265
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