Linear Operators: Spectral theory |
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Page 861
... inverse map T - 1 is continuous . To see that 7 is also continuous it will first be shown that T ( X ) is closed in B ( X ) . To do this the following criterion is useful : an element Te B ( X ) is in 7 ( X ) if and only if ( Ty ) z T ...
... inverse map T - 1 is continuous . To see that 7 is also continuous it will first be shown that T ( X ) is closed in B ( X ) . To do this the following criterion is useful : an element Te B ( X ) is in 7 ( X ) if and only if ( Ty ) z T ...
Page 877
... inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of 9 is the same as its spectrum as an element of X. = ( yy - 1 ) * == e * = PROOF . If y1 exists as an element of Y then , since X and Y ...
... inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of 9 is the same as its spectrum as an element of X. = ( yy - 1 ) * == e * = PROOF . If y1 exists as an element of Y then , since X and Y ...
Page 1311
... inverse . This convenient assumption is equivalent to the supposition that the operator 7 has been replaced by T - 2 . Our first result is concerned with the number k of linearly in- dependent boundary conditions which define T. Notice ...
... inverse . This convenient assumption is equivalent to the supposition that the operator 7 has been replaced by T - 2 . Our first result is concerned with the number k of linearly in- dependent boundary conditions which define T. Notice ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero