Linear Operators, Part 2 |
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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 1440
... sufficiently large , R ( tef , f ) ≥ 0 for fe D ( To ( T ) ) . If this is not the case , there exists a sequence { g } of elements of D ... sufficient- ly small subinterval [ a 。, b ) of 1440 XIII . ORDINARY DIFFERENTIAL OPERATORS XIII.7.9.
... sufficiently large , R ( tef , f ) ≥ 0 for fe D ( To ( T ) ) . If this is not the case , there exists a sequence { g } of elements of D ... sufficient- ly small subinterval [ a 。, b ) of 1440 XIII . ORDINARY DIFFERENTIAL OPERATORS XIII.7.9.
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero