Linear Operators, Part 2 |
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Page 1297
... norm is the norm of the pair [ f , T1f ] as an element of the graph of T1 ( t ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) D ( T1 ( t ) ) is complete in the norm f1 . Since the two additional terms in f2 are the ...
... norm is the norm of the pair [ f , T1f ] as an element of the graph of T1 ( t ) . Now T1 ( 7 ) is an adjoint ( Theorem 10 ) ; therefore ( cf. XII.1.6 ) D ( T1 ( t ) ) is complete in the norm f1 . Since the two additional terms in f2 are the ...
Page 1431
... norm of D ( T1 ( t ' ) ) coincides with the closure of D ( To ( t ' ) ) in the norm of D ( T1 ( t ) ) . 1 Let D1 and D , be the closures of D ( To ( t ' ) ) in the norms of D ( T1 ( t ' ) ) and D ( T1 ( 7 ) ) respectively . By step ( c ) ...
... norm of D ( T1 ( t ' ) ) coincides with the closure of D ( To ( t ' ) ) in the norm of D ( T1 ( t ) ) . 1 Let D1 and D , be the closures of D ( To ( t ' ) ) in the norms of D ( T1 ( t ' ) ) and D ( T1 ( 7 ) ) respectively . By step ( c ) ...
Page 1699
... norm of HP ) ( L ) of a sequence { g } of functions in Co ( L ) . Putting ĝ , ( x ) 0 for x in Ce - L , it follows from Definition 3.15 that Fe is the limit in the norm of H ( P ) ( C ) of the sequence { g ; } of elements of Co ( C ) ...
... norm of HP ) ( L ) of a sequence { g } of functions in Co ( L ) . Putting ĝ , ( x ) 0 for x in Ce - L , it follows from Definition 3.15 that Fe is the limit in the norm of H ( P ) ( C ) of the sequence { g ; } of elements of Co ( C ) ...
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