Linear Operators: Spectral theory |
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Page 1220
... fact that u ( e ) 0 and proves the theorem in case n = 1 . = Now suppose that the theorem is proved for n < p where p is a positive integer at most equal to the multiplicity of the ordered representation . By the inductive hypothesis ...
... fact that u ( e ) 0 and proves the theorem in case n = 1 . = Now suppose that the theorem is proved for n < p where p is a positive integer at most equal to the multiplicity of the ordered representation . By the inductive hypothesis ...
Page 1245
... fact that each complex number a has a unique representation a = reio , where r≥ 0 , and e1o | 1. By analogy with the fact that r = ( ăα ) 1 , we shall first seek to obtain the self adjoint operator A from the operator T * T . = α 1 ...
... fact that each complex number a has a unique representation a = reio , where r≥ 0 , and e1o | 1. By analogy with the fact that r = ( ăα ) 1 , we shall first seek to obtain the self adjoint operator A from the operator T * T . = α 1 ...
Page 1348
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of tσ = - λσ is ô1 ( t , λ ) = e - tv - λ , ô1⁄2 ( t , λ ) = et√ – à ...
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of tσ = - λσ is ô1 ( t , λ ) = e - tv - λ , ô1⁄2 ( t , λ ) = et√ – à ...
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