Linear Operators: Spectral theory |
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Page 1092
... finite number N of non - zero eigenvalues , we write λ „ ( T ) = 0 , n > N ) . Then , for each positive integer m ... finite- dimensional range , it is enough to prove the lemma in the special case that T has finite - dimensional domain ...
... finite number N of non - zero eigenvalues , we write λ „ ( T ) = 0 , n > N ) . Then , for each positive integer m ... finite- dimensional range , it is enough to prove the lemma in the special case that T has finite - dimensional domain ...
Page 1455
... finite below λ . 26 τ COROLLARY . Let t be a formally symmetric formal differential operator , and let T be any closed symmetric extension ( in particular , any self adjoint extension ) of TÔ ( t ) . Then τ is finite below 2 if and only ...
... finite below λ . 26 τ COROLLARY . Let t be a formally symmetric formal differential operator , and let T be any closed symmetric extension ( in particular , any self adjoint extension ) of TÔ ( t ) . Then τ is finite below 2 if and only ...
Page 1459
... finite below any finite λ . PROOF . We use the notations of the proof of Theorem 8. By Lemma 29 and Theorem 28 it is sufficient to show that t ' is finite below 20 in order to conclude that ▾ is finite below 2. But it was shown in the ...
... finite below any finite λ . PROOF . We use the notations of the proof of Theorem 8. By Lemma 29 and Theorem 28 it is sufficient to show that t ' is finite below 20 in order to conclude that ▾ is finite below 2. But it was shown in the ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero