## Linear Operators: Spectral operators |

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Page 2029

Similarly , functions on RN may

sometimes

is equivalent to the statement that for any two polynomials P , Q in N variables we

...

Similarly , functions on RN may

**determine**tempered distributions just as theysometimes

**determine**distributions . For example , since convergence 4n →g in Ởis equivalent to the statement that for any two polynomials P , Q in N variables we

...

Page 2054

( vi ) This uniquely

) o ( 0 ) = ( T - 1 exp ( tÂ ) ) * q 0 ) and the elements of the matrix representing the

convolution operator r - lexp ( tÂ ) are expressible in terms of a finite number of ...

( vi ) This uniquely

**determined**function o ( t ) is given by the equation o ( t ) = T ( t) o ( 0 ) = ( T - 1 exp ( tÂ ) ) * q 0 ) and the elements of the matrix representing the

convolution operator r - lexp ( tÂ ) are expressible in terms of a finite number of ...

Page 2373

which exclude the whole complex plane from the spectrum of the operator they

operators having eigenvalues 2 = sa

ks .

which exclude the whole complex plane from the spectrum of the operator they

**determine**. ... These boundary conditions**determine**discrete differentialoperators having eigenvalues 2 = sa

**determined**by equations of the form sin s =ks .

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algebra of projections analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear closure commuting compact complete consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero