Linear Operators: Spectral theory |
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Page 984
... follows that k * g = 0. Since 7 ( k + f * ƒ ) ( m ) > 0 for every m in M the operator T ( k + f * f ) is contained in no maximal ideal in A , and hence it follows from Lemma IX.1.12 ( e ) that it has an inverse al + T ( a ) in A1 . Thus ...
... follows that k * g = 0. Since 7 ( k + f * ƒ ) ( m ) > 0 for every m in M the operator T ( k + f * f ) is contained in no maximal ideal in A , and hence it follows from Lemma IX.1.12 ( e ) that it has an inverse al + T ( a ) in A1 . Thus ...
Page 993
... follows from what has just been demonstrated that xv , ay , i.e. , ap is independent of V. Q.E.D. = αγυνι = 16 ... follows from Lemma 3.6 that equation ( i ) holds for any open set with finite measure . It follows from the regularity of ...
... follows from what has just been demonstrated that xv , ay , i.e. , ap is independent of V. Q.E.D. = αγυνι = 16 ... follows from Lemma 3.6 that equation ( i ) holds for any open set with finite measure . It follows from the regularity of ...
Page 996
... follows from the above equation that f * 90 . From Lemma 12 ( b ) it is seen that o ( f * y ) Co ( y ) and from Lemma 12 ( c ) and the equation f Tf it follows that o ( ƒ * 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 ( f * y ) Co ( y ) and from Lemma 12 ( c ) and the equation f Tf it follows that o ( ƒ * q ) contains no interior point of o ( p ) . Hence o ( ƒ * q ) ...
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