Communications of the
Korean Mathematical Society CKMS

Hankel operators and their variants have abundant applications in numerous fields. For a non-zero complex number $r$, the $r$-Hankel operators on a Hilbert space $\mathcal{H}$ define a class of one such variant. This article introduces and explores some properties of two other variants of Hankel operators namely $k^{th}$-order $(C,r)$-Hankel operators and $k^{th}$-order $(R,r)$-Hankel operators $(k \geq 2)$ which are closely related to $r$-Hankel operators in such a way that a $k^{th}$-order $(C,r)$-Hankel matrix is formed from $r^k$-Hankel matrix on deleting every consecutive $(k-1)$ columns after the first column and a $k^{th}$-order $(R, r^k)$-Hankel matrix is formed from $r$-Hankel matrix if after the first column, every consecutive $(k-1)$ columns are deleted. For $|r| \neq 1$, the characterizations for the boundedness of these operators are also completely investigated. Finally, an appropriate approach is also presented to extend these matrices to two-way infinite matrices.

Keywords: Hilbert space, $r$-Hankel operator, $k^{th}$-order $(C,r)$-Hankel operator, $k^{th}$-order $(R,r)$-Hankel operator

MSC numbers: Primary 47B35; Secondary 47B02