Linear Operators: Spectral theory |
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Page 1190
... operator which is either symmetric or self adjoint according to the following definition . 7 DEFINITION . The operator T is said to be symmetric if = ( Tx , y ) ( x , Ty ) for every pair x , y of points in D ( T ) . It is said to be ...
... operator which is either symmetric or self adjoint according to the following definition . 7 DEFINITION . The operator T is said to be symmetric if = ( Tx , y ) ( x , Ty ) for every pair x , y of points in D ( T ) . It is said to be ...
Page 1223
... operator which will be studied in greater detail in the next chapter : the differential operator iD = i ( d / dt ) in the space L2 ( 0 , 1 ) . How are we to choose its ... symmetric then XII.4.1 1223 EXTENSIONS OF A SYMMETRIC TRANSFORMATION.
... operator which will be studied in greater detail in the next chapter : the differential operator iD = i ( d / dt ) in the space L2 ( 0 , 1 ) . How are we to choose its ... symmetric then XII.4.1 1223 EXTENSIONS OF A SYMMETRIC TRANSFORMATION.
Page 1270
... symmetric operators . The problem of determining whether a given symmetric operator has a self adjoint extension is of crucial importance in determining whether the spectral theorem may be employed . If the answer to this problem is ...
... symmetric operators . The problem of determining whether a given symmetric operator has a self adjoint extension is of crucial importance in determining whether the spectral theorem may be employed . If the answer to this problem is ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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