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.
In an arbitrary -algebra, a self-adjoint or hermitian element is any element such that .
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Walter Rudin, Functional analysis