Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

The development of novel LTA4H modulators to selectively target LTB4 generation

The pro-inflammatory mediator leukotriene B4 (LTB4) is implicated in the pathologies of an array of diseases and thus represents an attractive therapeutic target. The enzyme leukotriene A4 hydrolase (LTA4H) catalyses the distal step in LTB4 synthesis and hence inhibitors of this enzyme have been actively pursued. Despite potent LTA4H inhibitors entering clinical trials all have failed to show efficacy. We recently identified a secondary anti-inflammatory role for LTA4H in degrading the neutrophil chemoattractant Pro-Gly-Pro (PGP) and rationalized that the failure of conventional LTA4H inhibitors may be that they inadvertently prevented PGP degradation. We demonstrate that these inhibitors do indeed fail to discriminate between the dual activities of LTA4H, and enable PGP accumulation in mice. Accordingly, we have developed novel compounds that potently inhibit LTB4 generation whilst leaving PGP degradation unperturbed. These novel compounds could represent a safer and superior class of LTA4H inhibitors for translation into the clinic

Subdominant/Cryptic CD8 T Cell Epitopes Contribute to Resistance against Experimental Infection with a Human Protozoan Parasite

During adaptive immune response, pathogen-specific CD8+ T cells recognize preferentially a small number of epitopes, a phenomenon known as immunodominance. Its biological implications during natural or vaccine-induced immune responses are still unclear. Earlier, we have shown that during experimental infection, the human intracellular pathogen Trypanosoma cruzi restricts the repertoire of CD8+ T cells generating strong immunodominance. We hypothesized that this phenomenon could be a mechanism used by the parasite to reduce the breath and magnitude of the immune response, favoring parasitism, and thus that artificially broadening the T cell repertoire could favor the host. Here, we confirmed our previous observation by showing that CD8+ T cells of H-2a infected mice recognized a single epitope of an immunodominant antigen of the trans-sialidase super-family. In sharp contrast, CD8+ T cells from mice immunized with recombinant genetic vaccines (plasmid DNA and adenovirus) expressing this same T. cruzi antigen recognized, in addition to the immunodominant epitope, two other subdominant epitopes. This unexpected observation allowed us to test the protective role of the immune response to subdominant epitopes. This was accomplished by genetic vaccination of mice with mutated genes that did not express a functional immunodominant epitope. We found that these mice developed immune responses directed solely to the subdominant/cryptic CD8 T cell epitopes and a significant degree of protective immunity against infection mediated by CD8+ T cells. We concluded that artificially broadening the T cell repertoire contributes to host resistance against infection, a finding that has implications for the host-parasite relationship and vaccine development