Linear Operators: Spectral theory |
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Page 1050
... function f defined on Euclidean n - space E " , supposing that has a finite number of " singularities " at which it is not Lebesgue integrable , and defining a certain Cauchy - type principal value integral for f . 1 DEFINITION . Let f ...
... function f defined on Euclidean n - space E " , supposing that has a finite number of " singularities " at which it is not Lebesgue integrable , and defining a certain Cauchy - type principal value integral for f . 1 DEFINITION . Let f ...
Page 1196
... definition is as follows . 5 DEFINITION . Let E be the resolution of the identity for the self adjoint operator T and let ƒ be a complex Borel function defined E - almost everywhere on the real axis . Then the operator f ( T ) is defined ...
... definition is as follows . 5 DEFINITION . Let E be the resolution of the identity for the self adjoint operator T and let ƒ be a complex Borel function defined E - almost everywhere on the real axis . Then the operator f ( T ) is defined ...
Page 1645
... functions f such that of belongs to L2 ( E2 ) . ” In order for such an answer to make sense , it is desirable that we should be able to define a for every function , differentiable or not , and irrespective of whether aaf belongs to L2 ...
... functions f such that of belongs to L2 ( E2 ) . ” In order for such an answer to make sense , it is desirable that we should be able to define a for every function , differentiable or not , and irrespective of whether aaf belongs to L2 ...
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