Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 873
... set in M of all M , with 2 e 4. To see that M is dense in M suppose the contrary and let { M || x¿ ( M ) —x¿ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that 8 maps open sets onto open sets note ...
... set in M of all M , with 2 e 4. To see that M is dense in M suppose the contrary and let { M || x¿ ( M ) —x¿ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that 8 maps open sets onto open sets note ...
Page 993
... open set V. If f is in L1 ( R ) ~ L2 ( R ) , f vanishes on the complement of V , and f ( m ) = 1 for m in an open subset Vo of V , then the above proof shows that ( Pf ) ( m ) ay for every m in Vo , from which it follows that ¤Ã ̧ = ay ...
... open set V. If f is in L1 ( R ) ~ L2 ( R ) , f vanishes on the complement of V , and f ( m ) = 1 for m in an open subset Vo of V , then the above proof shows that ( Pf ) ( m ) ay for every m in Vo , from which it follows that ¤Ã ̧ = ay ...
Page 1151
... sets in R. We select an open set G1 in R such that FOKCG , G1 F2 = $ , and then choose an open set H1 such that F1⁄2 ○ K1 С H ̧ ‚ Ã ̧ˆ ( Ƒ1 ~ Ğ1 ) = þ . By induction , choose open sets G and H , such that F1KG , = n n 2 1 Ğ12 ( F2 H1 ...
... sets in R. We select an open set G1 in R such that FOKCG , G1 F2 = $ , and then choose an open set H1 such that F1⁄2 ○ K1 С H ̧ ‚ Ã ̧ˆ ( Ƒ1 ~ Ğ1 ) = þ . By induction , choose open sets G and H , such that F1KG , = n n 2 1 Ğ12 ( F2 H1 ...
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