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 1395
... range of the projection E ( σ1 ) contains the range of T. Choose a neighborhood V of 2 which is disjoint from o1 , and let † ( μ ) = ( λ — μ ) 1 if μ ‡ V and f ( u ) = 0 if u e V. Suppose that y is in the range of E ( o1 ) ; then by ...
... range of the projection E ( σ1 ) contains the range of T. Choose a neighborhood V of 2 which is disjoint from o1 , and let † ( μ ) = ( λ — μ ) 1 if μ ‡ V and f ( u ) = 0 if u e V. Suppose that y is in the range of E ( o1 ) ; then by ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero