Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm a bit confused about adjoint operators. Let $T:X \to Y$ be a linear isomorphism between Hilbert spaces. Then is it true that $(Tx,y)_Y = (x,T^*y)$ exists (does $T^*:Y \to X$ always exist)? What if these Hilbert spaces are part of Hilbert triples? Then what is the deal with the operator $T':Y^* \to X^*$?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

If you have a linear operator $T : X \to Y$ between the Hilbert spaces $X$ and $Y$, you have to distinguish between to notions of the dual operator.

  1. The Hilbert space adjoint: Define $T^* : Y \to X$ by $( T^*(y), x)_X = (y, T x)_Y$. Note that $x \mapsto (y, Tx)_Y$ is a linear functional on $X$ and, hence, can be identified with an element in $X$.

  2. The "usual" adjoint: Define $T^* : Y' \to X'$ by $\langle T^*(y^*), x \rangle_{X',X} = \langle y^*, T x\rangle_{Y',Y}$. Note that $x \mapsto \langle y,Tx \rangle_{X',X}$ is a bounded linear functional on $X$, hence an element of $X'$.

Here, $(\cdot,\cdot)_X$ refers to the scalar product in $X$, whereas $\langle \cdot, \cdot \rangle_{X',X}$ refers to the duality product between $X'$ and $X$.

Of course, both adjoints are linked via the Riesz isomorphisms of $X$ and $Y$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.