Adjoints techniques - ONSSA

duality come from? The short answer is: adjoints In fact adjoints play a key role in numerous 8 branches of engineering and computation but what exactly is an adjoint? This article describes how adjoints are used to compute sensitivities and then derives the adjoint of a linear time-10 varying system as an operator between Hilbert spaces

6 2 c J Fessler December 21 2004 13:3 (student version) 6 2 Fundamentals We write T : D → Y when T is a transformation from a set D in a vector space X to a vector space Y

Passage to adjoints is a very nice operation The map that sends ˝ to ˝ is conjugate linear and moreover the conjugate symmetry of the inner products shows that ˝ = ˝ for any linear operator There is a very close connection between adjunctions orthogonality and kernels and images Proposition 2

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C 6 Adjoints for Operators on a Hilbert Space 251 Hence D = D as an operator mapping S into L2(R) The important point in the preceding example is that if g 2 S is xed then f 7!hDf;gi is actually a bounded linear functional on S even though D is an unbounded operator The following exercise extends this to general operators Exercise C 49

We list and prove some useful properties of adjoints Proposition 2 Let Xbe a Hilbert space S : X!Xand T : X!Xbe bounded linear operators and ; 2F any two scalars We then have: i)( S+ T) = S + T ii)(ST) = T S iii)(T ) = T iv) kT k= kTk v) kTT k= kT Tk= kTk2 Proof i) and ii): Exercise iii) Fix any y2X We have hx;T yi= hTx;yi= hy;Txi = hTy