Title: Quasinormality and weak quasinormality of operators
Abstract: There are two notions to define the quasinormality of unbounded operators by Kaufman and Stochel-Szafraniec respectively. Our results show that Kaufman’s definition of an unbounded quasinormal operator coincides with that given by Stochel-Szafraniec. In this talk we discuss various characterizations of unbounded quasinormal operators. Examples demonstrating the sharpness of our results are constructed. An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. This approach establishes a new definition to be called weakly quasinormal operators. Some characterizations concerning to the weakly quasinormal operators are discussed. In addition, various examples and counterexamples illustrating the concepts of this work are constructed by using weighted shifts on directed trees.
Title: Weighted shifts on directed trees
Abstract: The main goal of this research is to implement some methods of graph theory into operator theory. We do it by introducing a new class of Hilbert space operators, which we propose to call weighted shifts on directed trees. We have studied the structure of these operators since 2012 and obtained some remarkable results about those operators. In this talk we discuss some of the established properties for such operators, for examples, normality, hyponormality, and subnormality, etc. It is well-known that the subnormality of Hilbert space operators is closely related to Stieltjes moment sequences. We consider directed trees with one branching vertex and establish a connection between the subnormality and Stieltjes moment sequences by using the k-step backward extendability of bounded subnormal weighted shifts. In addition, some exotic examples of weighted shifts on directed trees related to Stieltjes moment sequences are discussed.
Title: Beyond Standard Scores: Using Item-level Responses From Clinical Measures to Detect Atypical Developmental Patterns
Abstract: Developmental deviance (DDEV) refers to the non-sequential attainment of milestones within a developmental domain. This observation is in contrast to developmental delay (DD), where milestones are reached in the typical sequence, but the timeline of attainments is delayed. There is evidence that DDEV is associated with certain neurobehavioral diagnoses, such as autism spectrum disorder (ASD). Clinically, the attainment of developmental milestones is assessed through standardized measures of developmental domains. Many psychometric tests are arranged hierarchically, and on the surface, two individuals with the same overall score on a clinical measure may appear to be impaired to a similar extent. However, at the itemized level, individuals with DDEV exhibit a more scattered pattern of incorrect answers. Differentiating between DDEV and DD may inform prognosis and predict long-term outcomes. We developed a measure of scatter, called deviance index (DI), to differentiate between DD and DDEV using standardized measures of language ability. We tested the accuracy of DI to predict ASD diagnosis, and by extension DDEV, among individuals from the New Jersey Language and Autism Genetics Study (NJLAGS) cohort. Using our DI metric, we found that individuals with ASD and a language impairment (LI) exhibit more DDEV across measures of expressive, pragmatic, and metalinguistic language compared to individuals with LI alone. By distinguishing between DDEV and DD, DI was able to predict ASD diagnosis among LI/LI+ASD probands. DI can be applied to measures across multiple developmental domains in order to characterize developmental profiles of individuals with DDEV/DD.