## Linear Operators: Spectral theory |

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A bounded operator T in Hilbert space s is called

A bounded operator T in Hilbert space s is called

**unitary**if TT * = T * T = 1 ; it is called self adjoint , symmetric T ; or Hermitian if T = T * ; positive if it is self adjoint and if ( Tx , x ) 0 for every x in V ; and positive ...Page 1146

R = R1 • ii 1 --- i = 0 Rm , and say that R is the direct sum of R2 , ... , Rn . The following theorem is easily proved by induction in case R is

R = R1 • ii 1 --- i = 0 Rm , and say that R is the direct sum of R2 , ... , Rn . The following theorem is easily proved by induction in case R is

**unitary**, and thus follows in the general case by the theorem stated above . THEOREM .Page 1148

Moreover , H is the direct sum of a finite number of groups Hi , each of which is either ( 1 ) The additive group of the real axis ; or ( 2 ) The group SU ( n ) of all n xn complex

Moreover , H is the direct sum of a finite number of groups Hi , each of which is either ( 1 ) The additive group of the real axis ; or ( 2 ) The group SU ( n ) of all n xn complex

**unitary**matrices of determinant l ; or ( 3 ) The group ...### What people are saying - Write a review

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extensive presentation of applications of the spectral theorem | 911 |

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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero