Can the adjoint of unbounded operators bounded?

Question

Can the adjoint of an unbounded operator be bounded? If not, how to show it? Examples are appreciated.

For instance, given an unbounded operator $V: \mathcal{K} \otimes \mathcal{H} \to \mathcal{K} $, where $\mathcal{K}$ and $\mathcal{H}$ are Hilbert spaces, can its adjoint be bounded?

Answer 1

on a Banach space $X$, the adjoint of $T: X \to X$ is an operator $S : X^* \to X^*$ such that for every $Y \subseteq X$ onto which $T$ is bounded, for every $y \in Y^*, x \in Y$ : $$y(Tx) = (Sy)(x)$$

if $T$ is unbounded then there exists $\|y\| =\|x\| = 1$ and $x_n \to x$ such that $$y(Tx_n) \to \infty$$ hence $$(Sy)(x_n) \to \infty$$