  This research extends quantum singular value transformation (QSVT) for
general bounded operators embedded in unitary operators on possibly
infinite-dimensional Hilbert spaces. Through in-depth mathematical exploration,
we have achieved a refined operator-theoretic understanding of QSVT, leading to
a more streamlined approach. One of the key discoveries is that polynomial
transformations in QSVT inherently apply to the entire operator, rather than
being contingent on the selection of a specific basis. We expect that this
research will pave the way for applying these insights to a broader range of
problems in quantum information processing and provide analytical tools for
quantum dynamics, such as quantum walks.
