Linear Operators: Spectral theory |
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Page 1247
... unique positive " square root " . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 Ꭲ . = == PROOF . By Lemma 2 , o ( T ) C [ 0 , ∞ ) and , by Theorem 2.6 ...
... unique positive " square root " . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 Ꭲ . = == PROOF . By Lemma 2 , o ( T ) C [ 0 , ∞ ) and , by Theorem 2.6 ...
Page 1250
... unique . By Lemma 1.6 ( c ) , AP * = T * . Hence T * T = AP * PA . Since , by Lemma 5 , P * P is a projection onto R ( 4 ) , it follows that T * T = 42 . The uniqueness of A now follows from Lemma 3. Since A is unique , P is uniquely ...
... unique . By Lemma 1.6 ( c ) , AP * = T * . Hence T * T = AP * PA . Since , by Lemma 5 , P * P is a projection onto R ( 4 ) , it follows that T * T = 42 . The uniqueness of A now follows from Lemma 3. Since A is unique , P is uniquely ...
Page 1283
... unique solution ( cf. Lemma VII.3.4 ) ∞ F = ( 1 + Φ ) = 1 Η = Σ ( -1 ) ΦΗ . j = 0 Since all the terms in equation ( e ) but the first are absolutely contin- uous , it follows that F is absolutely continuous . Thus Theorem 1 is proved ...
... unique solution ( cf. Lemma VII.3.4 ) ∞ F = ( 1 + Φ ) = 1 Η = Σ ( -1 ) ΦΗ . j = 0 Since all the terms in equation ( e ) but the first are absolutely contin- uous , it follows that F is absolutely continuous . Thus Theorem 1 is proved ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 deficiency indices Definition denote dense 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 linear 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 transform unique unitary vanishes vector zero