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In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.
An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of ) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all.
6 Notes. 7 References. ... an element of a *-algebra is called self-adjoint if it is the same as its adjoint ... Operator Algebras. Theory of C*-Algebras and von ...
Linear Operators is a three-volume textbook on the theory of linear operators, written by Nelson Dunford and Jacob T. Schwartz. The three volumes are (I) General Theory; (II) Spectral Theory, Self Adjoint Operators in Hilbert Space; and (III) Spectral Operators. The first volume was published in 1958, the second in 1963, and the third in 1971.
Example.Multiplication by a non-negative function on an L 2 space is a non-negative self-adjoint operator.. Example.Let U be an open set in R n.On L 2 (U) we consider differential operators of the form
A compact operator K on H is symmetrizable if there is a bounded self-adjoint operator S on H such that S is positive with trivial kernel, i.e. (Sx,x) > 0 for all non-zero x, and SK is self-adjoint: =. In many applications S is also compact. The operator S defines a new inner product on H
That is, if is a compact self-adjoint operator on , then the essential spectra of and that of + coincide, i.e. () = (+). This explains why it is called the essential spectrum : Weyl (1910) originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions.
on its domain. When A is a self-adjoint operator =, in the sense of the spectral theorem and this notation is used more generally in semigroup theory. The cogenerator of the semigroup is the contraction defined by = (+) ().