Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1190
... everywhere defined operator then the statements T * 2 T and T * T are equivalent and thus a bounded operator is symmetric if and only if it is self adjoint . If T is an everywhere defined sym- metric operator then T * 2T and thus T * T ...
... everywhere defined operator then the statements T * 2 T and T * T are equivalent and thus a bounded operator is symmetric if and only if it is self adjoint . If T is an everywhere defined sym- metric operator then T * 2T and thus T * T ...
Page 1212
... everywhere on e . Consequently ( B „ f ) ( 2 ) = √§ ̧ ƒ ( 8 ) Wn + 1 ( 8 , A ) y ( ds ) , fЄ L1 ( Sn , v ) , λεεη n ... everywhere in S , and since USS this equality must hold v - almost everywhere in S. Q.E.D. 10 DEFINITION . Let W be ...
... everywhere on e . Consequently ( B „ f ) ( 2 ) = √§ ̧ ƒ ( 8 ) Wn + 1 ( 8 , A ) y ( ds ) , fЄ L1 ( Sn , v ) , λεεη n ... everywhere in S , and since USS this equality must hold v - almost everywhere in S. Q.E.D. 10 DEFINITION . Let W be ...
Page 1233
... everywhere defined , bounded operator of norm not more than ( 2 ) . Consequently , the series ∞ Σ ( 2—20 ) " R ( 2 ) * + 1 n = 0 converges if 2-201 < ( 26 ) . Since T , is closed , we have = ∞ 1 ( T1 — λ1 ) Σ ( 2−2 。) ” R ( 2 ...
... everywhere defined , bounded operator of norm not more than ( 2 ) . Consequently , the series ∞ Σ ( 2—20 ) " R ( 2 ) * + 1 n = 0 converges if 2-201 < ( 26 ) . Since T , is closed , we have = ∞ 1 ( T1 — λ1 ) Σ ( 2−2 。) ” R ( 2 ...
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