Linear Operators, Part 2 |
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Page 1540
... essential spectrum of 7 coincides with the essential spectrum of the operator T1 ( t ) + B . All Let be a regular formal differential operator on an interval I , and let B be a linear operator in L2 ( I ) defined in D ( T1 ( t ) ) which ...
... essential spectrum of 7 coincides with the essential spectrum of the operator T1 ( t ) + B . All Let be a regular formal differential operator on an interval I , and let B be a linear operator in L2 ( I ) defined in D ( T1 ( t ) ) which ...
Page 1595
... spectrum are the following : ( 9 ) Under the assumptions ( a ) and ( b ) of ( 5 ) , and with the further assumption that for all x [ \ p ( t ) g ( t ) \ -1/2 dt = ∞ , the essential spectrum of 7 is the entire real axis . ( 6.21 ) ...
... spectrum are the following : ( 9 ) Under the assumptions ( a ) and ( b ) of ( 5 ) , and with the further assumption that for all x [ \ p ( t ) g ( t ) \ -1/2 dt = ∞ , the essential spectrum of 7 is the entire real axis . ( 6.21 ) ...
Page 1596
... essential spectrum of 7 is smaller than the least point in the essential spectrum of 7 + r . This assertion is not true if it is not assumed that 7 is bounded below ( cf. the exercises in Section 9.D ) . τ Additional criteria are given ...
... essential spectrum of 7 is smaller than the least point in the essential spectrum of 7 + r . This assertion is not true if it is not assumed that 7 is bounded below ( cf. the exercises in Section 9.D ) . τ Additional criteria are given ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero