If a linear operator takes an orthonormal basis to an orthonormal set, then is the orthonormal set a basis?

Question

Let $H$ be a Hilbert space and $T:H\rightarrow H$ be a bounded linear operator which takes an orthonormal basis $x_i$ to an orthonormal set $y_i$, i.e., $y_i=Tx_i$ for all $i$. Then does $y_i$ form a basis? This is trivial for finite dimensions, but what about infinite-dimensions?

Answer 1

No. There is an obvious counterexample for infinite dimensions.

Hint: Is every injective map from the positive integers to the positive integers also a bijection?

Let $Tx_i = x_{i+1}$.

Answer 2

No, not necessarily. Consider the operator $T:\ell^2\to\ell^2$ that satisfies $$ T e_i = e_{2i} $$ for all $i\in\mathbb N$, where $\{e_i\}_{i\in\mathbb N}$ is an orthonormal basis.