Linear Operators, Part 2 |
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Page 992
... Lemma 3.6 ( i ) that μ ( V ) is finite and thus , as was observed in the note following the proof of that lemma ... preceding lemma that Pf , does likewise . It will next be shown that Øf is constant on Un . To see this , let m1 , m2 ...
... Lemma 3.6 ( i ) that μ ( V ) is finite and thus , as was observed in the note following the proof of that lemma ... preceding lemma that Pf , does likewise . It will next be shown that Øf is constant on Un . To see this , let m1 , m2 ...
Page 1192
... preceding lemma shows that ( × I — T ) −1 exists as a bounded operator . To prove that a is in p ( T ) it will therefore suffice to prove that its domain is closed and has orthocomplement zero . Since T T it follows from Lemma 1.6 ( a ) ...
... preceding lemma shows that ( × I — T ) −1 exists as a bounded operator . To prove that a is in p ( T ) it will therefore suffice to prove that its domain is closed and has orthocomplement zero . Since T T it follows from Lemma 1.6 ( a ) ...
Page 1474
... LEMMA . If 2 is in J and 21 is in Jn + 1 , then λ < λ1 . n PROOF . Suppose this is false . Then 1 < λ . Since , by ... preceding Lemma 41 , 0 < ( 2 — ¿ ‚ ) [ * œ ( t , λ ) o ( t , ¿ , ) dt = få { ( xo ) ( t , 2 ) o ( t , 2 ) —o ( t , 21 ) ...
... LEMMA . If 2 is in J and 21 is in Jn + 1 , then λ < λ1 . n PROOF . Suppose this is false . Then 1 < λ . Since , by ... preceding Lemma 41 , 0 < ( 2 — ¿ ‚ ) [ * œ ( t , λ ) o ( t , ¿ , ) dt = få { ( xo ) ( t , 2 ) o ( t , 2 ) —o ( t , 21 ) ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 deficiency indices Definition denote dense domain eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero