Let H be an infinite dimensional separable complex Hilbert space with an orthonormal basis \((e_ n)\). Let End(H) be the Banach algebra of endomorphisms of H. To an operator \(A\in End(H)\) we associate a matrix \((a_{ij})\) defined by \(a_{ij}=<Ae_ i\), \(e_ j>\), i,j\(\geq 1\). If \(d_ k(A)=_{| i-j| =k}| a_{ij}|\), \(k\in Z\), the set of all integers, and assuming that for some weight \(\alpha\) from Z to \(R^+=\{t\in R:\) \(t>0\}\), we have \(\sum_{k\in Z}d_ k(A)\alpha (k)<\infty,\) the author proves the following:
Theorem: If A is left (right) invertible and one of the following conditions is satisfied:
1. \(k^{-1}\ln \alpha (k)\) converges to 0, as \(| k|\) increases to \(\infty.\)
2. There is \(q>1\), \(c>0:\) \(d_ k(A)\leq c(1+| k|^{-q}).\)
3. There is \(\epsilon >0\), \(c>0:\alpha\) (k)\(\geq c\) exp(\(\epsilon\) k), \(k\in Z.\)
Then there exists a left (right) inverse operator \(B\in End(H)\), whose matrix satisfies the corresponding condition:
1’. \(\sum_{k\in Z}d_ k(B)\alpha (k)<\infty;\)
2’. \(d_ k(B)\leq c_ 1\) \((| k| +1)^{-q};\)
3’. \(d_ k(B)\leq c_ 1\exp (-\epsilon_ 0k)\), \(k\in Z\), for some \(\epsilon_ 0>0\).