Rota’s universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In articular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Towards the Knowledge-based Design of Universal Influenza Epitope Ensemble Vaccines

Motivation: Influenza A viral heterogeneity remains a significant threat due to unpredictable antigenic drift in seasonal influenza and antigenic shifts caused by the emergence of novel subtypes. Annual review of multivalent influenza vaccines targets strains of influenza A and B likely to be predominant in future influenza seasons. This does not induce broad, cross protective immunity against emergent subtypes. Better strategies are needed to prevent future pandemics. Cross-protection can be achieved by activating CD8+ and CD4+ T cells against highly-conserved regions of the influenza genome. We combine available experimental data with informatics-based immunological predictions to help design vaccines potentially able to induce cross-protective T-cells against multiple influenza subtypes. Results: To exemplify our approach we designed two epitope ensemble vaccines comprising highlyconserved and experimentally-verified immunogenic influenza A epitopes as putative non-seasonal influenza vaccines; one specifically targets the US population and the other is a universal vaccine. The USA-specific vaccine comprised 6 CD8+ T cell epitopes (GILGFVFTL, FMYSDFHFI, GMDPRMCSL, SVKEKDMTK, FYIQMCTEL, DTVNRTHQY) and 3 CD4+ epitopes (KGILGFVFTLTVPSE, EYIMKGVYINTALLN, ILGFVFTLTVPSERG). The universal vaccine comprised 8 CD8+ epitopes: (FMYSDFHFI, GILGFVFTL, ILRGSVAHK, FYIQMCTEL, ILKGKFQTA, YYLEKANKI, VSDGGPNLY, YSHGTGTGY) and the same 3 CD4+ epitopes. Our USA-specific vaccine has a population protection coverage (portion of the population potentially responsive to one or more component epitopes of the vaccine, PPC) of over 96% and 95% coverage of observed influenza subtypes. The universal vaccine has a PPC value of over 97% and 88% coverage of observed subtypes.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.