Linear Operators: Spectral theory |
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Page 891
... function E on a field of subsets of an abstract set S. The functions we shall integrate are the bounded L - measurable functions . A Z - measurable function ( cf. IV.2.12 ) is a function f with f1 ( A ) e Σ for every Borel set A in the ...
... function E on a field of subsets of an abstract set S. The functions we shall integrate are the bounded L - measurable functions . A Z - measurable function ( cf. IV.2.12 ) is a function f with f1 ( A ) e Σ for every Borel set A in the ...
Page 893
... measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral ƒ ƒ ( s ) E ( ds ) where E is merely a bounded additive operator valued set function , we observe that the ...
... measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral ƒ ƒ ( s ) E ( ds ) where E is merely a bounded additive operator valued set function , we observe that the ...
Page 900
... measurable scalar functions on S determined by the bounded E - measurable scalar functions on S in such a way that each equivalence class consists of all E - measurable functions which differ from some bounded E - measurable function ...
... measurable scalar functions on S determined by the bounded E - measurable scalar functions on S in such a way that each equivalence class consists of all E - measurable functions which differ from some bounded E - measurable function ...
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