Multiplicative Preservers and Induced Operators

Dedicated to Professor Graciano de Oliveira on the occasion of his retirement. Let V be an n-dimensional Hilbert space. Suppose H is a subgroup of the symmetric group of degree m, and χ: H → C is a character of degree 1 on H. Consider the symmetrizer on the tensor space ⊗ m V S(v1 ⊗ · · · ⊗ vm) = 1 |H| χ(σ)vσ−1 (1) ⊗ · · · ⊗ vσ−1 (m) σ∈H defined by H and χ. The subspace V m χ (H) of ⊗ m V spanned by S( ⊗ m V) is called the symmetry class of tensors over V associated with H and χ. The elements in V m χ (H) of the form S(v1 ⊗ · · · ⊗ vm) are called decomposable tensors and are denoted by v1 ∗ · · · ∗ vm. For any linear operator T acting on V, there is an (unique) induced operator Kχ(T) (or just K(T) for notational simplicity) acting on V m χ (H) satisfying K(T)v1 ∗... ∗ vm = T v1 ∗ · · · ∗ T vm. We characterize multiplicative maps φ such that F (φ(T)) = F (T) for all operators T acting on V, where F are various scalar or set valued functions including the spectral radius, (decomposable) numerical radius, spectral norm, spectrum, (decomposable) numerical range of T or K(T)

Newborn screening for severe combined immunodeficiency in 11 screening programs in the United States.

ImportanceNewborn screening for severe combined immunodeficiency (SCID) using assays to detect T-cell receptor excision circles (TRECs) began in Wisconsin in 2008, and SCID was added to the national recommended uniform panel for newborn screened disorders in 2010. Currently 23 states, the District of Columbia, and the Navajo Nation conduct population-wide newborn screening for SCID. The incidence of SCID is estimated at 1 in 100,000 births.ObjectivesTo present data from a spectrum of SCID newborn screening programs, establish population-based incidence for SCID and other conditions with T-cell lymphopenia, and document early institution of effective treatments.DesignEpidemiological and retrospective observational study.SettingRepresentatives in states conducting SCID newborn screening were invited to submit their SCID screening algorithms, test performance data, and deidentified clinical and laboratory information regarding infants screened and cases with nonnormal results. Infants born from the start of each participating program from January 2008 through the most recent evaluable date prior to July 2013 were included. Representatives from 10 states plus the Navajo Area Indian Health Service contributed data from 3,030,083 newborns screened with a TREC test.Main outcomes and measuresInfants with SCID and other diagnoses of T-cell lymphopenia were classified. Incidence and, where possible, etiologies were determined. Interventions and survival were tracked.ResultsScreening detected 52 cases of typical SCID, leaky SCID, and Omenn syndrome, affecting 1 in 58,000 infants (95% CI, 1/46,000-1/80,000). Survival of SCID-affected infants through their diagnosis and immune reconstitution was 87% (45/52), 92% (45/49) for infants who received transplantation, enzyme replacement, and/or gene therapy. Additional interventions for SCID and non-SCID T-cell lymphopenia included immunoglobulin infusions, preventive antibiotics, and avoidance of live vaccines. Variations in definitions and follow-up practices influenced the rates of detection of non-SCID T-cell lymphopenia.Conclusions and relevanceNewborn screening in 11 programs in the United States identified SCID in 1 in 58,000 infants, with high survival. The usefulness of detection of non-SCID T-cell lymphopenias by the same screening remains to be determined