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When the self-adjoint operator in question is compact, this version of the spectral theorem reduces to something similar to the finite-dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.
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.
A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert ...
Part of the statement of the theorem is that this derivative exists—i.e., that is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since U t {\displaystyle U_{t}} is only assumed (ahead of time) to be continuous, and not differentiable.
In particular, for self-adjoint operators, the spectrum lies on the real line and (in general) is a spectral combination of a point spectrum of discrete eigenvalues and a continuous spectrum. [ 15 ] Spectral theory briefly
If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators.Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum.
In the classical case T was a compact self-adjoint operator; in this case T is just a self-adjoint bounded operator with 0 ≤ T ≤ I. The abstract theory of spectral measure can therefore be applied to T to give the eigenfunction expansion for D. The central idea in the proof of Weyl and Kodaira can be explained informally as follows.