Linear Operators: Spectral theory |
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Page 1297
... called stronger than a set C , ( f ) = 0 , j linear combination of the B ,. Two sets of boundary conditions are called equivalent if each is stronger than the other . A complete set of boundary values is a maximal linearly independent ...
... called stronger than a set C , ( f ) = 0 , j linear combination of the B ,. Two sets of boundary conditions are called equivalent if each is stronger than the other . A complete set of boundary values is a maximal linearly independent ...
Page 1432
... called the order of the singularity of equation [ * ] at zero . If v = 0 , there is no singularity at all , and zero is called a regular point of the differential equation . If v = 1 , the singularity of equation [ * ] at zero is called ...
... called the order of the singularity of equation [ * ] at zero . If v = 0 , there is no singularity at all , and zero is called a regular point of the differential equation . If v = 1 , the singularity of equation [ * ] at zero is called ...
Page 1451
... called a bound for T , and the smallest ( largest ) such c is called the upper ( lower ) bound for T. If ( Tx , x ) ≥ 0 for all x in D ( T ) , then T is called non- negative . 20 DEFINITION . Let be a formally symmetric formal dif ...
... called a bound for T , and the smallest ( largest ) such c is called the upper ( lower ) bound for T. If ( Tx , x ) ≥ 0 for all x in D ( T ) , then T is called non- negative . 20 DEFINITION . Let be a formally symmetric formal dif ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero