Correction to: One-sided invertibility of discrete operators with bounded coefficients

1 Correction to: Aequat. Math. https://doi.org/10.1007/s00010-020-00773-8

2 Addendum

By the sufficiency in Theorem 3.12, the left (resp., right) invertibility of the operators $A\in \big \{A_E,\widetilde{A}_E,A_{E_m},\widetilde{A}_{E_m}\big \}$ on the spaces $l^p$ ($1<p<\infty$) follows from the invertibility of the operator $A^\diamond A$ (resp., $AA^\diamond$). The Fredholmness of such operators A is not necessary for their one-sided invertibility (for example, the right invertible operator $A_{E_m}= E_m$ is not Fredholm for $|m|>1$). On the other hand, the invertibility criterion for the operators $B\in \{A^\diamond A,AA^\diamond \}$ presented in Theorem 3.15 is also valid if $B\in {\mathcal W}_p$ and $A\notin {\mathcal W}_p$.

As a result, Theorems 4.4, 4.5, 5.4, 5.5 can be essentially reinforced by excluding the Fredholm condition for the operators $A_E,\widetilde{A}_E,A_{E_m},\widetilde{A}_{E_m}$, respectively, and applying the invertibility criterion from Theorem 3.15.

The modified theorems have the following form.

Theorem 6.1

The operator $A_E$ given by (4.2) and satisfying (4.3) is left invertible on the space $l^p$ for $p\in (1,\infty )$ if for $B=A_E^\diamond A_E$ and all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ defined for $B=A_E^\diamond A_E$ by (3.33) is invertible. The operator $A_E$ given by (4.2) and satisfying (4.3) and condition (A) is right invertible on the space $l^p$ for $p\in (1,\infty )$ if for $B=A_EA_E^\diamond$ and all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ defined for $B=A_EA_E^\diamond$ by (3.33) is invertible. Under these conditions, one of the left (resp., right) inverses of the operator $A_E$ is given by $A_E^L=(A_E^\diamond A_E)^{-1}A_E^\diamond$ (resp., by $A_E^R=A_E^\diamond (A_EA_E^\diamond )^{-1}$).

Theorem 6.2

The operator $\widetilde{A}_E$ given by (4.2) and satisfying (4.3) is left invertible on the space $l^p$ for $p\in (1,\infty )$ if for $B=\widetilde{A}_E^\diamond \widetilde{A}_E$ and all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ defined for $B=\widetilde{A}_E^\diamond \widetilde{A}_E$ by (3.33) is invertible. The operator $\widetilde{A}_E$ given by (4.2) and satisfying (4.3) and condition (B) is right invertible on the space $l^p$ for $p\in (1,\infty )$ if E is a permutation operator and for $B=\widetilde{A}_E\widetilde{A}_E^\diamond$ and for all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ defined for $B=\widetilde{A}_E\widetilde{A}_E^\diamond$ by (3.33) is invertible. Under these conditions, one of the left (resp., right) inverses of the operator $\widetilde{A}_E$ is given by $\widetilde{A}_E^L= (\widetilde{A}_E^\diamond \widetilde{A}_E)^{-1}\widetilde{A}_E^\diamond$ (resp., by $\widetilde{A}_E^R=\widetilde{A}_E^\diamond (\widetilde{A}_E\widetilde{A}_E^\diamond )^{-1}$).

Theorem 6.3

Let $p\in (1,\infty )$ and $m\in \mathbb {Z}\setminus \{0\}$. Then the slant-dominated discrete Wiener-type operator $A_{E_m}$ is left (resp., right) invertible on the space $l^p$ if for $B=A_{E_m}^\diamond A_{E_m}$ (resp., for $B= A_{E_m}A_{E_m}^\diamond$) and for all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ given by (3.33) for $B=A_{E_m}^\diamond A_{E_m}$ (resp., for $B=A_{E_m}A_{E_m}^\diamond$) is invertible. Under these conditions, one of the left (resp., right) inverses of the operator $A_{E_m}$ is given by $A_{E_m}^L=(A_{E_m}^\diamond A_{E_m})^{-1} A_{E_m}^\diamond$ (resp., by $A_{E_m}^R=A_{E_m}^\diamond (A_{E_m} A_{E_m}^\diamond )^{-1}$).

Theorem 6.4

Let $p\in (1,\infty )$ and $m\in \mathbb {Z}\setminus \{0\}$. Then the slant-dominated discrete Wiener-type operator $\widetilde{A}_{E_m}$ is left (resp., right) invertible on the space $l^p$ if for $B=\widetilde{A}_{E_m}^\diamond \widetilde{A}_{E_m}$ (resp., for $B=\widetilde{A}_{E_m}\widetilde{A}_{E_m}^\diamond$) and for all sufficiently large $n\in \mathbb {N}$ the operators $B_n^\pm =P_n^\pm BP_n^\pm$ are invertible on the spaces $P_n^\pm l^p$, respectively, and the $(2n-1)\times (2n-1)$ matrix $B_{n,0}$ given by (3.33) for $B=\widetilde{A}_{E_m}^\diamond \widetilde{A}_{E_m}$ (resp., for $B=\widetilde{A}_{E_m} \widetilde{A}_{E_m}^\diamond$) is invertible, where $|m|=1$ in the case of right invertibility. Under these conditions, one of the left (resp., right) inverses of the operator $\widetilde{A}_{E_m}$ is given by $\widetilde{A}_{E_m}^L= (\widetilde{A}_{E_m}^\diamond \widetilde{A}_{E_m})^{-1}\widetilde{A}_{E_m}^\diamond$ (resp., by $\widetilde{A}_{E_m}^R=\widetilde{A}_{E_m}^\diamond (\widetilde{A}_{E_m}\widetilde{A}_{E_m}^\diamond )^{-1}$).