Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1010
... norm or the double - norm of T. The class of all Hilbert - Schmidt operators on will be denoted by HS . In this definition of the class HS a particular orthonormal sequence was used . The following lemma shows that the class HS depends ...
... norm or the double - norm of T. The class of all Hilbert - Schmidt operators on will be denoted by HS . In this definition of the class HS a particular orthonormal sequence was used . The following lemma shows that the class HS depends ...
Page 1015
... norm of HS it follows from Lemma VII.6.5 that the contour C of the integral in [ * ] contains σ ( T ) for all suffi- ciently large n . From Corollary VII.6.3 it is seen that , in the norm of HS + , lim [ 2 , -T2 - 1 = [ 2 , −T ] −1 n1 ...
... norm of HS it follows from Lemma VII.6.5 that the contour C of the integral in [ * ] contains σ ( T ) for all suffi- ciently large n . From Corollary VII.6.3 it is seen that , in the norm of HS + , lim [ 2 , -T2 - 1 = [ 2 , −T ] −1 n1 ...
Page 1297
... norm is the norm of the pair [ ƒTƒ ] as an element of the graph of T1 ( τ ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) ( T , ( t ) ) is complete in the norm f1 . Since the two additional terms in [ f2 are the ...
... norm is the norm of the pair [ ƒTƒ ] as an element of the graph of T1 ( τ ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) ( T , ( t ) ) is complete in the norm f1 . Since the two additional terms in [ f2 are the ...
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 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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF 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