Linear Operators: Spectral theory |
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Page 1017
... basis { y1 , ... , Yn } . This proves the first statement . Q.E.D. We recall that the characteristic polynomial of an operator A in En is found by representing A as a matrix with respect to any convenient basis for E " and calculating ...
... basis { y1 , ... , Yn } . This proves the first statement . Q.E.D. We recall that the characteristic polynomial of an operator A in En is found by representing A as a matrix with respect to any convenient basis for E " and calculating ...
Page 1028
... basis for H. Since E is finite dimensional we may suppose without loss of generality that there is a finite subset B of A such that { x , x = B } is an orthonormal basis for E§ , and { x , α = A − B } is an orthonormal basis for ( I ...
... basis for H. Since E is finite dimensional we may suppose without loss of generality that there is a finite subset B of A such that { x , x = B } is an orthonormal basis for E§ , and { x , α = A − B } is an orthonormal basis for ( I ...
Page 1029
... basis { 1 , ... , -1 } for S with ( ( T — ÎI ) x ̧ , x , ) = 0 for j > i . Let x be orthogonal to S and have norm one so that { x1 , ... , x } is an orthonormal basis for E " . Then the matrix of T - II in terms of { x1 , . . . , x } is ...
... basis { 1 , ... , -1 } for S with ( ( T — ÎI ) x ̧ , x , ) = 0 for j > i . Let x be orthogonal to S and have norm one so that { x1 , ... , x } is an orthonormal basis for E " . Then the matrix of T - II in terms of { x1 , . . . , x } is ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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(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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero