Linear Operators: Spectral theory |
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Page 890
... scalar functions f to which the formula may be applied . One class of scalar functions f , other than polynomials , for which the operator f ( T ) has already been defined is the class C ( o ( T ) ) of all complex continuous functions ...
... scalar functions f to which the formula may be applied . One class of scalar functions f , other than polynomials , for which the operator f ( T ) has already been defined is the class C ( o ( T ) ) of all complex continuous functions ...
Page 1178
... scalar - valued functions into functions with values in l . It is plain from Plancherel's theorem that is a bounded mapping of the space L2 of scalar - valued functions into the space L2 ( 2 ) of square - integrable vector - valued ...
... scalar - valued functions into functions with values in l . It is plain from Plancherel's theorem that is a bounded mapping of the space L2 of scalar - valued functions into the space L2 ( 2 ) of square - integrable vector - valued ...
Page 1782
... scalar product n . . . , n ( iv ) ( [ x1 , ... , x „ ] , [ Y1 , · · · , Yn ] ) = Σ ( X¡ , Yi ) v i = 1 i where ( • , • ) , is the scalar product in X ,. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To ...
... scalar product n . . . , n ( iv ) ( [ x1 , ... , x „ ] , [ Y1 , · · · , Yn ] ) = Σ ( X¡ , Yi ) v i = 1 i where ( • , • ) , is the scalar product in X ,. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero