Linear Operators: Spectral theory |
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Page 1092
... range , it is enough to prove the lemma in the special case that T has finite - dimensional domain and range . Note that if T has finite - dimensional range , T = ET , where E is the orthogonal projection on the range of T. Thus T * = T ...
... range , it is enough to prove the lemma in the special case that T has finite - dimensional domain and range . Note that if T has finite - dimensional range , T = ET , where E is the orthogonal projection on the range of T. Thus T * = T ...
Page 1134
... range of each projection E , in- variant , and the set F of projections E , λe C , subdiagonalizes T. To prove the second proposition of our theorem , we have only to verify that if E is any other orthogonal projection such that E ≥ Ex ...
... range of each projection E , in- variant , and the set F of projections E , λe C , subdiagonalizes T. To prove the second proposition of our theorem , we have only to verify that if E is any other orthogonal projection such that E ≥ Ex ...
Page 1397
... range R ( T1 ) of T1 coincides with the range of T and is therefore closed . Moreover , the orthocomplement of R ( T ) is N ; ( cf. XII.1.6 ) hence [ * ] 2 $ 5 = R ( T ) N = R ( T1 ) → N. 2 Suppose now that T2 is a proper symmetric ...
... range R ( T1 ) of T1 coincides with the range of T and is therefore closed . Moreover , the orthocomplement of R ( T ) is N ; ( cf. XII.1.6 ) hence [ * ] 2 $ 5 = R ( T ) N = R ( T1 ) → N. 2 Suppose now that T2 is a proper symmetric ...
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