Linear Operators: Spectral theory |
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Page 984
... follows that k✶ g = 0. Since τ ( k + f ⋆ fƒ ) ( m ) > 0 k * for every m in M the operator T ( k + f * f ) is contained in no maximal ideal in A1 and hence it follows from Lemma IX.1.12 ( e ) that it has an inverse al + T ( a ) in A1 ...
... follows that k✶ g = 0. Since τ ( k + f ⋆ fƒ ) ( m ) > 0 k * for every m in M the operator T ( k + f * f ) is contained in no maximal ideal in A1 and hence it follows from Lemma IX.1.12 ( e ) that it has an inverse al + T ( a ) in A1 ...
Page 993
... follows from what has just been demonstrated that xv , = αvov1 = αv , i.e. , ap is independent of V. αγυνι ay ... follows from Lemma 3.6 that equation ( i ) holds for any open set with finite measure . It follows from the regularity of μ ...
... follows from what has just been demonstrated that xv , = αvov1 = αv , i.e. , ap is independent of V. αγυνι ay ... follows from Lemma 3.6 that equation ( i ) holds for any open set with finite measure . It follows from the regularity of μ ...
Page 996
... follows from the above equation that f * 90 . From Lemma 12 ( b ) it is seen that o ( fy ) Co ( q ) and from Lemma 12 ( c ) and the equation tƒ = tf it follows that o ( f * q ) contains no interior point of o ( p ) . Hence o ( ƒ * q ) ...
... follows from the above equation that f * 90 . From Lemma 12 ( b ) it is seen that o ( fy ) Co ( q ) and from Lemma 12 ( c ) and the equation tƒ = tf it follows that o ( f * q ) contains no interior point of o ( p ) . Hence o ( ƒ * q ) ...
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