Linear Operators: Spectral theory |
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Page 1436
... converges , then { f } converges . Let { g } be a bounded sequence of elements of D ( T ) such that { Tgn } converges . Find a subsequence { n , } { h } such that x ( h1 ) converges for each j , 1 ≤ j ≤k . Then h1 = h ̧ — Σ ^ _ ̧x ...
... converges , then { f } converges . Let { g } be a bounded sequence of elements of D ( T ) such that { Tgn } converges . Find a subsequence { n , } { h } such that x ( h1 ) converges for each j , 1 ≤ j ≤k . Then h1 = h ̧ — Σ ^ _ ̧x ...
Page 1461
... converges . The mapping To ( t + 71 ) is defined and linear on the finite dimensional space N ,, hence continuous on this space . Consequently , { ( t + t1 + μ + ε ) gm } converges . Consequently , putting Īm = 1m − 8m , { Īm } is a ...
... converges . The mapping To ( t + 71 ) is defined and linear on the finite dimensional space N ,, hence continuous on this space . Consequently , { ( t + t1 + μ + ε ) gm } converges . Consequently , putting Īm = 1m − 8m , { Īm } is a ...
Page 1664
... converges unconditionally to F. PROOF . It follows from the Definition 37 of the topology in D ( C ) that it suffices to show that ( 27 ) - " Σ Γι C ei L · x q ( x ) dx converges unconditionally to F ( p ) for each 9 in Ca ( C ) . For ...
... converges unconditionally to F. PROOF . It follows from the Definition 37 of the topology in D ( C ) that it suffices to show that ( 27 ) - " Σ Γι C ei L · x q ( x ) dx converges unconditionally to F ( p ) for each 9 in Ca ( C ) . For ...
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