Fourier Series, Fourier Transform and Their Applications to Mathematical Physics by Valery Serov

Author:Valery Serov
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

with some constant . We assume without loss of generality that . This condition implies that exists, it is defined over all of H, and . Indeed, exists and is bounded because if and only if . The norm estimate follows from

Since is bounded, is a closed subspace in H. The self-adjointness of A means that . Therefore, is closed and , i.e., is densely defined. Therefore, and . Since

we may conclude in this case that , , and

where is the spectral family of . Let us note that and , which follows from the spectral theorem for bounded operators and from the fact that if and only if . Next, let us define the operator , , as

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