Linear Operators: Spectral theory |
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Page 1015
... norm of HS it follows from Lemma VII.6.5 that the contour C of the integral in [ * ] contains σ ( T ) for all suffi- ciently large n . From Corollary VII.6.3 it is seen that , in the norm of HS + , lim [ 2 , T1 = [ 2 , −T ] -1 n44 ...
... norm of HS it follows from Lemma VII.6.5 that the contour C of the integral in [ * ] contains σ ( T ) for all suffi- ciently large n . From Corollary VII.6.3 it is seen that , in the norm of HS + , lim [ 2 , T1 = [ 2 , −T ] -1 n44 ...
Page 1297
... norm of ƒ as an element of H " ( J ) it follows easily that D ( T1 ( T ) ) is also complete under the norm | ƒ2 . As f1 ≤ƒ , it follows from Theorem II.2.5 that the two norms are equivalent . The lemma follows immediately from this ...
... norm of ƒ as an element of H " ( J ) it follows easily that D ( T1 ( T ) ) is also complete under the norm | ƒ2 . As f1 ≤ƒ , it follows from Theorem II.2.5 that the two norms are equivalent . The lemma follows immediately from this ...
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 form ...
... 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 form ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero