Linear Operators: Spectral theory |
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Page 1088
... operator in L2 ( S , E , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { μ , } an enumeration of the correspond- ing eigenvalues . Show that if g = Kf for some f in ... Compact Operators Generalized Carleman Inequalities.
... operator in L2 ( S , E , μ ) . Let { 9 } be an enumeration of the eigenfunctions of K , and { μ , } an enumeration of the correspond- ing eigenvalues . Show that if g = Kf for some f in ... Compact Operators Generalized Carleman Inequalities.
Page 1089
... compact or non- compact operator satisfy the inequalities Ꭲ Un + m + 1 ( Ti + T2 ) Sun + 1 ( Ti ) + um + 1 ( T2 ) Mn + m + 1 ( TT2 ) = n + 1 ( Ti ) um + 1 ( T , ) . PROOF . We observe that max min , Pn + XI.9.1 1089 CLASSES OF COMPACT ...
... compact or non- compact operator satisfy the inequalities Ꭲ Un + m + 1 ( Ti + T2 ) Sun + 1 ( Ti ) + um + 1 ( T2 ) Mn + m + 1 ( TT2 ) = n + 1 ( Ti ) um + 1 ( T , ) . PROOF . We observe that max min , Pn + XI.9.1 1089 CLASSES OF COMPACT ...
Page 1095
... operators such that T - Tmp → 0 as m , n → ∞ , there exists a compact operator T such that TT ( in the topology of C1 ) as n∞∞ . PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform ...
... operators such that T - Tmp → 0 as m , n → ∞ , there exists a compact operator T such that TT ( in the topology of C1 ) as n∞∞ . PROOF . By Lemma 9 ( a ) and the fact that the family of compact operators is closed in the uniform ...
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