Toeplitz Operators Whose Symbols Have Zeros

Let X be a topological linear space and let the operator A ∈ L(X) = L (X,X) have a nonclosed image (im A ≠ clos im A). Since a linear operator is in fact a triplet (correspondence rule, domain of definition, domain of values), its properties depend in an essential manner on each of the components of the triplet and there are reasons to hope that by appropriately perturbing these components one can achieve that the image of the “new” (perturbed) operator will be closed. This problem is intimately connected with problems that arise when solving the equation Ax = y.If we are interested in describing all its right-hand sides for which the solution lies in a given space, then we fix the domain of definition of the operator and try to restrict its domain of values, simultaneously modifying the topology in the latter. If, however, we are interested in describing all “reasonable” solutions for a given class of right-hand sides, then the domain of values of the operators is kept fixed, while the operator itself and its domain of definition are subject to a certain extension. The “new” operator with closed image, as well and the procedure for constructing it, are called a normalization of the operator A.