Linear Operators, Part 2 |
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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 Ex variant , and the set F of projections E , EC , 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 EE or EE for ...
... range of each projection Ex variant , and the set F of projections E , EC , 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 EE or EE for ...
Page 1395
... range of the projection E ( o1 ) contains the range of T. ‡ 01 , Choose a neighborhood V of 2 which is disjoint from σ1 , and let f ( u ) = ( 2 - μ ) -1 if u V and f ( u ) = 0 if u e V. Suppose that y is in the range of E ( o1 ) ; then ...
... range of the projection E ( o1 ) contains the range of T. ‡ 01 , Choose a neighborhood V of 2 which is disjoint from σ1 , and let f ( u ) = ( 2 - μ ) -1 if u V and f ( u ) = 0 if u e V. Suppose that y is in the range of E ( o1 ) ; then ...
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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF 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