Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1708
... proof of Lemma 3 is complete . Q.E.D. PROOF ( OF THEOREM 2 ) . Let J be a domain whose closure is contained in I. Then , by Lemmas 4.13 and 3.13 and by Definition 3.15 , fJ is in A ( n ) ( J ) for some n . Since by Lemma 3.18 and Lemma ...
... proof of Lemma 3 is complete . Q.E.D. PROOF ( OF THEOREM 2 ) . Let J be a domain whose closure is contained in I. Then , by Lemmas 4.13 and 3.13 and by Definition 3.15 , fJ is in A ( n ) ( J ) for some n . Since by Lemma 3.18 and Lemma ...
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 are ...
... 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 are ...
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
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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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 compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain 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 isometric isomorphism kernel L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero