Let ▫$\mathscr{H}$ ▫be a complex separable Hilbert space, let ▫$\mathscr{B(H)}$▫ denote the algebra of all bounded linear operators on ▫$\mathscr{H}$▫ and let ▫$k$▫ be a positive integer. Given a sequence of nonnegative integers ▫$r_1 \ge r_2 \ge \dots \ge 0$▫ we show that there exists a subspace ▫$\mathcal{S} \subseteq \mathscr{B(H)}$▫, such that its ▫$k$▫-reflexivity defect is equal to ▫$r_k$▫ for all ▫$k \ge 1$▫. For a finite dimensional complex Hilbert space we give an explicit formula for the reflexivity defect of the kernel of an arbitrary elementary operator of length 2, i.e., an operator, acting on the algebra ▫$\mathscr{B(H)}$▫, of the form ▫$\Delta(T) = A_1TB_1 - A_2TB_2$▫ where ▫$A_1$▫, ▫$A_2$▫ and ▫$B_1$, $B_2$▫ are given pairs of linearly independent operators. We characterize the ▫$k$▫-reflexivity defect of the image of a generalized derivation. Using the latter we also give an explicit formula for the ▫$k$▫-reflexivity defect of the image of an elementary operator on ▫ $\mathscr{B(H)}$▫ of the form ▫$\Delta(T) = ATB - T$▫ where ▫$A, \: B \in \mathscr{B(H)}$▫ are given operators. We also consider the ▫$k$▫-reflexivity and the $k$-hyperreflexivity of some subspaces of operators over the orthogonal direct sum of complex separable Hilbert spaces. We give a lower and upper bound for the $k$-hyperreflexivity constant of such a space and we prove that the lower bound is optimal. Furthermore, we give similar estimates when the direct sum of Hilbert spaces is not necessary orthogonal. In this case the bounds for the ▫$k$▫-hyperreflexivity constant depend also on the angles between the given Hilbert spaces. We also consider the hyperreflexivity constant of the low dimensional algebras of matrices that have a noncommutative lattice of invariant subspaces.
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