Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certain apriori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such as $n$- and $d$-normality, the (generalized and ordinary) Fredholm property, and appropriate apriori estimates.