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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.
Download QR code; Print/export ... an element of a *-algebra is called self-adjoint if it is the same as its ... Operator Algebras. Theory of C*-Algebras and von ...
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.
E is called self-adjoint if E(B) is self-adjoint for all B. E is called spectral if it is self-adjoint and () = () for all ,. We will assume throughout that E is regular. Let C(X) denote the abelian C*-algebra of continuous functions on X.
The essential spectrum is a subset of the spectrum σ, and its complement is called the discrete spectrum, so = ().If T is self-adjoint, then, by definition, a number λ is in the discrete spectrum of T if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define B ≥ A {\displaystyle B\geq A} if the following hold: A {\displaystyle A} and B {\displaystyle B} are self-adjoint
In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator.It is named after the mathematician Kurt Friedrichs.
Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let : denote the adjoint of .