Linear Operators: Spectral theory |
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Page 1226
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
Page 1696
... lemma , let m be a sequence in C ( D ) with Fm G in the topology of H ( * ) ( D ) as m → ∞ . Using Lemma 2.1 , let the function y in Co ( I ) be such that y ( x ) = 1 for all in a neighborhood of C. Then , by Lemma 3.10 and Definition ...
... lemma , let m be a sequence in C ( D ) with Fm G in the topology of H ( * ) ( D ) as m → ∞ . Using Lemma 2.1 , let the function y in Co ( I ) be such that y ( x ) = 1 for all in a neighborhood of C. Then , by Lemma 3.10 and Definition ...
Page 1733
... lemma follows , as has been shown above . Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . σ 19 LEMMA ...
... lemma follows , as has been shown above . Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . σ 19 LEMMA ...
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