Linear Operators: Spectral theory |
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Page 1216
... null set N ,. Hence for every λ not in the μ - null set N = U1 N ; we have ( Tƒ ;, W ( " ) ( · , 2 ) ) = λ ( ƒ ;, W ( " ) ( • , 2 ) ) . Since the vectors [ ƒ ,, Tf ; ] are dense in the graph à ( " ) of T ( " ) we have a a ( T ( n ) f ...
... null set N ,. Hence for every λ not in the μ - null set N = U1 N ; we have ( Tƒ ;, W ( " ) ( · , 2 ) ) = λ ( ƒ ;, W ( " ) ( • , 2 ) ) . Since the vectors [ ƒ ,, Tf ; ] are dense in the graph à ( " ) of T ( " ) we have a a ( T ( n ) f ...
Page 1343
... set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m , are continuous . Thus it suffices to prove that one can construct the functions and Pi on any measurable ...
... set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m , are continuous . Thus it suffices to prove that one can construct the functions and Pi on any measurable ...
Page 1914
... set of all functions , III.2.1 ( 102 ) Milman . ( See Krein - Milman theorem ) Minimax principle , X.4 ( 908 ) ... Null function . ( See also Null set ) criterion for , III.6.8 ( 147 ) definition , III.2.3 ( 103 ) Null set . ( See also ...
... set of all functions , III.2.1 ( 102 ) Milman . ( See Krein - Milman theorem ) Minimax principle , X.4 ( 908 ) ... Null function . ( See also Null set ) criterion for , III.6.8 ( 147 ) definition , III.2.3 ( 103 ) Null set . ( See also ...
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