Linear Operators: Spectral theory |
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Page 1099
... sufficient to consider the case in which the Hilbert space is finite - dimensional . The argument in this special case is as follows . Since both sides . of ( 1 ) are continuous in T and since every finite matrix may be ap- proximated ...
... sufficient to consider the case in which the Hilbert space is finite - dimensional . The argument in this special case is as follows . Since both sides . of ( 1 ) are continuous in T and since every finite matrix may be ap- proximated ...
Page 1114
... sufficient to prove our theorem in the special case in which T has a finite - dimensional range . Ar- guments like those of the third paragraph of the proof of Lemma 6 then show that it is sufficient to prove our theorem for the case of ...
... sufficient to prove our theorem in the special case in which T has a finite - dimensional range . Ar- guments like those of the third paragraph of the proof of Lemma 6 then show that it is sufficient to prove our theorem for the case of ...
Page 1403
... sufficient to show that each λ € △ has a neighborhood 40 such that W , ( , λ ) e L2 ( a , c ) for μ - almost all λ € Д , since Д may then be written as a countable union of such neighborhoods 4. We shall show below that for each eД ...
... sufficient to show that each λ € △ has a neighborhood 40 such that W , ( , λ ) e L2 ( a , c ) for μ - almost all λ € Д , since Д may then be written as a countable union of such neighborhoods 4. We shall show below that for each eД ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero