Linear Operators: Spectral theory |
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Page 1459
... 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 proof of Theorem 8 that c ...
... 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 proof of Theorem 8 that c ...
Page 1724
... PROOF . By the preceding lemma and by Corollary 11 it suffices to show that ( Tf , g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green's formula , proved in the last paragraph of Section 2 , this equation is valid if ƒ and g ...
... PROOF . By the preceding lemma and by Corollary 11 it suffices to show that ( Tf , g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green's formula , proved in the last paragraph of Section 2 , this equation is valid if ƒ and g ...
Page 1750
... proof of Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2. The theorem is false if no ... PROOF ( of Theorem 1 ) . The proof will be given in a series of steps , some of which will be proofs of auxiliary ...
... proof of Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2. The theorem is false if no ... PROOF ( of Theorem 1 ) . The proof will be given in a series of steps , some of which will be proofs of auxiliary ...
Contents
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