Linear Operators, Part 2 |
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Page 1016
... basis { x } used to define it . It is quite evident that the trace is a linear function of A. 9 LEMMA . The trace of an operator in En is independent of the basis used to define it . In addition , if A and B are any two operators in En ...
... basis { x } used to define it . It is quite evident that the trace is a linear function of A. 9 LEMMA . The trace of an operator in En is independent of the basis used to define it . In addition , if A and B are any two operators in En ...
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 1029
... basis { ~ 1 , ... , -1 } for S with ( ( T - I ) 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 — ÎI in terms of { x1 , . . . , x „ } is ...
... basis { ~ 1 , ... , -1 } for S with ( ( T - I ) 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 — ÎI 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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero