The canonical $\boldsymbol{K}$-dual Bessel sequence of a $\boldsymbol{K}$-frame

Abstract

It is well known that the canonical $K$-dual Bessel sequence is the $K$-dual Bessel sequence whose analysis operator has minimal operator norm in all the $K$-dual Bessel sequences. However, we cannot directly know the specific form of the canonical $K$-dual Bessel sequence from this definition. In this paper, we first give the specific form of the canonical $K$-dual Bessel sequence and the optimal $K$-frame bounds of a $K$-frame for two special cases. Specially, the optimal $K$-frame bounds of a $K$-frame in the finite dimensional Hilbert space can be expressed by the eigenvalues. Lastly, the specific form of the canonical $K$-dual Bessel sequence that we obtain in this paper is used to characterize all the $K$-dual Bessel sequences.