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 ( σ ( 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 ( σ ( 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 L ( 12 ) 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 L ( 12 ) of square - integrable vector - valued ...
Page 1782
... scalar product n n ( iv ) ( [ x1 , ... , x , ] , [ Y1 , · · · , Yn ] ) = Σ ( x¿ , Yi ) i i • i = 1 where ( • , • ) , is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize ...
... scalar product n n ( iv ) ( [ x1 , ... , x , ] , [ Y1 , · · · , Yn ] ) = Σ ( x¿ , Yi ) i i • i = 1 where ( • , • ) , is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize ...
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