Linear Operators: Spectral theory |
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Page 1297
... norm is the norm of the pair [ ƒ , T1f ] as an element of the graph of T1 ( 7 ) . 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 [ ƒ , T1f ] as an element of the graph of T1 ( 7 ) . 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 ( 7 ' ) ) coincides with the closure of D ( To ( t ' ) ) in the norm of D ( T1 ( t ) ) . Let D1 and D2 be the closures of D ( To ( 7 ' ) ) in the norms of D ( T1 ( t ' ) ) and D ( T1 ( t ) ) respectively . By step ( c ) ...
... norm of D ( T1 ( 7 ' ) ) coincides with the closure of D ( To ( t ' ) ) in the norm of D ( T1 ( t ) ) . Let D1 and D2 be the closures of D ( To ( 7 ' ) ) in the norms of D ( T1 ( t ' ) ) and D ( T1 ( t ) ) respectively . By step ( c ) ...
Page 1639
... norm equivalent to the norm displayed , though not under the norm displayed itself . It is in the sense of these norms that we speak of the topology of C * ( I ) , C * ( I ) , etc. If is a formal partial differential operator of the ...
... norm equivalent to the norm displayed , though not under the norm displayed itself . It is in the sense of these norms that we speak of the topology of C * ( I ) , C * ( I ) , etc. If is a formal partial differential operator of the ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero