Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle\_U \ge \langle Tx, Ty\rangle\_V.$$
All unitary transformations satisfy this criterion; is there a larger class of linear transformations that do?



Which linear transformations between f.d. Hilbert spaces contract the inner product?
Such a map will preserve orthogonality, and any such map must be a scalar multiple of an isometry. This is true in great generality, e.g. the map $T$ doesn't have to be linear, and $U$ and $V$ don't have to be finite-dimensional; see Theorem 1 in



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> Chmieliński, *Linear mappings approximately preserving orthogonality.* J. Math. Anal. Appl. **304** (2005), no. 1, 158–169.
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Consequently, a map $T \colon U \to V$ contracts the inner product if and only if $T = \alpha S$, where S is an isometry and $|\alpha| \leq 1$.




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**Edit:** Here's a simple proof of the assertion that an orthogonality-preserving linear map between finite-dimensional inner product spaces is a scalar multiple of an isometry. Let $T \colon U \to V$ be such a map and fix an orthonormal basis $\{e\_1, \ldots, e\_n\}$ for $U$. Observe that $e\_i + e\_j \perp e\_i - e\_j$. Thus
$$ 0 = \langle T(e\_i + e\_j), T(e\_i - e\_j) \rangle = \langle Te\_i, Te\_i \rangle - \langle Te\_j, Te\_j \rangle. $$
So we may set $\alpha = \langle Te\_i, Te\_i \rangle$; this is a nonnegative constant independent of $i$. In particular, if $T$ kills one $e\_i$, it kills all the others. It follows that either $T=0$ or else $\{Te\_1, \ldots, Te\_n\}$ is an orthogonal basis for the range of $T$. In the latter situation, an easy computation yields
$$ \|Tx\|^2 = \sum\_i \frac{|\langle Tx, Te\_i \rangle|^2}{\langle Te\_i, Te\_i \rangle} = \sum\_i \frac{|\langle \sum\_j \langle x, e\_j \rangle Te\_j, Te\_i \rangle|^2}{\langle Te\_i, Te\_i \rangle} = \sum\_i |\langle x,e\_i \rangle|^2 \langle Te\_i, Te\_i \rangle = \alpha \|x\|^2 $$
for all $x \in U$. It follows that $\frac{1}{\sqrt{\alpha}}T$ is an isometry.


