Linear Operators: Spectral theory |
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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 , -7 , ] = [ 2 , −T ] −1 ...
... 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 , -7 , ] = [ 2 , −T ] −1 ...
Page 1297
... norm is the norm of the pair [ ƒ , T1ƒ ] as an element of the graph of T1 ( 7 ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) D ( T1 ( 7 ) ) is complete in the norm f1 . Since the two additional terms in f1⁄2 ...
... norm is the norm of the pair [ ƒ , T1ƒ ] as an element of the graph of T1 ( 7 ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) D ( T1 ( 7 ) ) is complete in the norm f1 . Since the two additional terms in f1⁄2 ...
Page 1782
... norm | ,, i.e. , if each of the spaces X is a normed linear space , then the space X is a normed linear space . The norm in X may be introduced in a variety of ways ; for example , any one of the following norms defines the product ...
... norm | ,, i.e. , if each of the spaces X is a normed linear space , then the space X is a normed linear space . The norm in X may be introduced in a variety of ways ; for example , any one of the following norms defines the product ...
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