Two methods for avoiding hereditary completeness

Hereditary completeness for systems of exponentials and reproducing kernels

“…In fact, first examples go back to Hamburger [13] who constructed a compact operator with a complete set of eigenvectors, whose restriction to an invariant subspace is a nonzero Volterra operator (and, hence, is not complete). Further examples of nonhereditarily complete systems were found by Markus [18] and Nikolski [19], while a general approach to constructing nonhereditarily complete systems was developed by Dovbysh, Nikolski and Sudakov [9,10]. Any nonhereditarily complete system gives an example of an exact system which is not a summation basis.…”

Spectral synthesis for exponentials and logarithmic length

“…The results of Sections 4 and 5 were obtained with the support of Russian Foundation for Basic Research grant 20-51-14001-ANF-a. aspects of abstract hereditary complete systems were considered in [9,10] while in [1,12] some interesting relations with operator algebras can be found.…”

Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators

“…A general approach to constructing nonhereditarily complete systems was developed by Dovbysh, Nikolski and Sudakov [9,10], where it was shown, in particular, that for any condition of closeness to an orthogonal basis (weaker than the quadratical closeness) there are uniformly minimal nonhereditarily complete systems that meet this condition. We recall that a complete minimal system that is quadratically close to an orthogonal basis is always a Riesz basis, by a classical theorem of Bari.…”