The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topol...The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.展开更多
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram...We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.展开更多
文摘The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.
文摘We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.
