Let $H$ be a Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we study the class of operators $A\in B(H)$ with closed range such that $AA^{+}-A^{+}A$ is invertible, where $A^{+}$ is the Moore-Penrose inverse of $A$. Also, we present new relations between $(AA^{*}+A^{*}A)^{-1}$ and $(A+A^{*})^{-1}$. The present paper is an extension of results from [J. Benítez and V. Rakočević, Appl. Math. Comput. 217 (2010) 3493-3503] to infinite-dimensional Hilbert space.