Linear Operators: Spectral theory |
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Page 891
... function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of ... measurable functions . A E - measurable function ( cf. IV.2.12 ) is a function ƒ with ƒ ̃1 ( 4 ) € Σ for every Borel ...
... function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of ... measurable functions . A E - measurable function ( cf. IV.2.12 ) is a function ƒ with ƒ ̃1 ( 4 ) € Σ for every Borel ...
Page 893
... measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral Sf ( s ) E ( ds ) where E is merely a bounded additive operator valued set function ... function on E , the integral ...
... measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral Sf ( s ) E ( ds ) where E is merely a bounded additive operator valued set function ... function on E , the integral ...
Page 900
... measurable then fo is a bounded 2 - measurable function , i.e. , an element of the B * -algebra B ( S , E ) . The ... measurable scalar functions on S determined by the bounded E - measurable scalar functions on S in such a way that each ...
... measurable then fo is a bounded 2 - measurable function , i.e. , an element of the B * -algebra B ( S , E ) . The ... measurable scalar functions on S determined by the bounded E - measurable scalar functions on S in such a way that each ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero