Linear Operators: Spectral operators |
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Page 993
... follows that av . 0 = = with compact closure . Then it follows from what has just been demonstrated that av , avuvi ay , i.e. , ap is independent of V. Q.E.D. = = 16 THEOREM . If the bounded measurable function has its spectral set ...
... follows that av . 0 = = with compact closure . Then it follows from what has just been demonstrated that av , avuvi ay , i.e. , ap is independent of V. Q.E.D. = = 16 THEOREM . If the bounded measurable function has its spectral set ...
Page 996
... follows from the above equation that f * q0 . From Lemma 12 ( b ) it is seen that o ( f * q ) Co ( 9 ) and from Lemma 12 ( c ) and the equation Tƒ = Tf it follows that o ( f * q ) contains no interior point of o ( q ) . Hence σ ( ƒ * q ) ...
... follows from the above equation that f * q0 . From Lemma 12 ( b ) it is seen that o ( f * q ) Co ( 9 ) and from Lemma 12 ( c ) and the equation Tƒ = Tf it follows that o ( f * q ) contains no interior point of o ( q ) . Hence σ ( ƒ * q ) ...
Page 1708
... follows that there exists some F in H ( + ) such that ( T1 + 0 ) F = ge . However , since fe is in Hm + P - 1 ) , and since by ( 5 ) , ( T1 + σε ) fεP = §ɛ , it follows that fɛ = F_is_in_H ( m + ” ) ( C ) so that a fortiori , fɛç is in ...
... follows that there exists some F in H ( + ) such that ( T1 + 0 ) F = ge . However , since fe is in Hm + P - 1 ) , and since by ( 5 ) , ( T1 + σε ) fεP = §ɛ , it follows that fɛ = F_is_in_H ( m + ” ) ( C ) so that a fortiori , fɛç is in ...
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 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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero