Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let if-abstract0/img1.gif" ALT="$ S$"> be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space if-abstract0/img2.gif" ALT="$ \mathfrak{H}$">, let if-abstract0/img3.gif" ALT="$ \mathcal{H}$"> be an auxiliary Hilbert space, let

if-abstract0/img4.gif" ALT="$\displaystyle J_\mathfrak{H}=\begin{pmatrix}0&-iI_\mathfrak{H} iI_\mathfrak{H} & 0\end{pmatrix}, $">

and let if-abstract0/img5.gif" ALT="$ J_\mathcal{H}$"> be defined analogously. A unitary relation if-abstract0/img6.gif" ALT="$ \Gamma$"> from the Krein space if-abstract0/img7.gif" ALT="$ (\mathfrak{H}^2,J_\mathfrak{H})$"> to the Krein space if-abstract0/img8.gif" ALT="$ (\mathcal{H}^2,J_\mathcal{H})$"> is called a boundary relation for the adjoint if-abstract0/img9.gif" ALT="$ S^*$"> if if-abstract0/img10.gif" ALT="$ \ker \Gamma=S$">. The corresponding Weyl family if-abstract0/img11.gif" ALT="$ M(\lambda)$"> is defined as the family of images of the defect subspaces if-abstract0/img12.gif" ALT="$ \widehat{\mathfrak{N}}_\lambda$">, if-abstract0/img13.gif" ALT="$ \lambda\in \mathbb{C}\setminus\mathbb{R}$">, under if-abstract0/img14.gif" ALT="$ \Gamma$">. Here if-abstract0/img15.gif" ALT="$ \Gamma$"> need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space if-abstract0/img16.gif" ALT="$ \mathcal{H}$"> and the class of unitary relations if-abstract0/img17.gif" ALT="$ \Gamma:(\mathfrak{H}^2,J_\mathfrak{H})\to(\mathcal{H}^2,J_\mathcal{H})$">, it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every if-abstract0/img18.gif" ALT="$ \mathcal{H}$">-valued maximal dissipative (for if-abstract0/img19.gif" ALT="$ \lambda\in\mathbb{C}_+$">) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

     

Time-changed Poisson processes of order k

Abstract

In this article, we study the Poisson process of order ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/lsaa20/2020/lsaa20.v038.i01/07362994.2019.1653198/20191202/images/lsaa_a_1653198_ilm0003.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lsaa20/2020/lsaa20.v038.i01/07362994.2019.1653198/20191202/images/lsaa_a_1653198_ilm0003.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />inline">i>ki> (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call, respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.     

A supplement to precise asymptotics in the law of the iterated logarithm

Let <img height="14" border="0" style="vertical-align:bottom" width="101" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X04006493-si1.gif"> be a sequence of real-valued i.i.d. random variables with E(X)=0 and E(X²)=1, and set <img height="18" border="0" style="vertical-align:bottom" width="92" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X04006493-si4.gif">, n?1. This paper studies the precise asymptotics in the law of the iterated logarithm. For example, using a result on convergence rates for probabilities of moderate deviations for <img height="14" border="0" style="vertical-align:bottom" width="78" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X04006493-si6.gif"> obtained by Li et al. [Internat. J. Math. Math. Sci. 15 (1992) 481-497], we prove that, for every b&isin;(&minus;1/2,1], iv class="formula" id=">iv class="mathml"><img height="80" border="0" style="vertical-align:bottom" width="432" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X04006493-si8.gif">     

An SQP method for minimization of locally Lipschitz functions with nonlinear constraints

Abstract

In this paper, we present a quadratic model for minimizing problems with nonconvex and nonsmooth objective and constraint functions. This method is based on sequential quadratic programming that uses an ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gopt20/2019/gopt20.v068.i04/02331934.2018.1545123/20190407/images/gopt_a_1545123_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gopt20/2019/gopt20.v068.i04/02331934.2018.1545123/20190407/images/gopt_a_1545123_ilm0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>li>1 penalty function to equilibrate among the decrease of the objective function and the feasibility of the constraints. To construct a quadratic subproblem, we linearize the objective and constraint functions with their <i>εi>-subdifferential approximations. These approximations are iteratively improved until an effective descent direction is found. Also, we prove that our method is globally convergent in the sense that, every accumulation point of the generated sequence is a Clark-stationary point for the penalty function. Finally, the presented algorithm is implemented in Matlab environment and compared with some recent methods.     

Maps preserving -Lie product on factor von Neumann algebras

Let ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0004.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0004.gif"}" /> and ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0005.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0005.gif"}" /> be two factor von Neumann algebras with dimensions greater than 1. In this paper, we consider a map ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0006.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0006.gif"}" /> from ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0007.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0007.gif"}" /> onto ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0008.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0008.gif"}" /> preserving the ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0009.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0009.gif"}" />-Lie product, that is, ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0010.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0010.gif"}" /> for all ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0011.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0011.gif"}" />. It is proved that a bijective map ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0012.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0012.gif"}" /> from ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0013.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0013.gif"}" /> onto ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0014.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0014.gif"}" /> preserves the ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0015.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0015.gif"}" />-Lie product, if and only if ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0016.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0016.gif"}" /> is a linear ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0017.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0017.gif"}" />-isomorphism or a conjugate linear ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0018.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0018.gif"}" />-isomorphism. In particular, if the factor von Neumann algebras ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0019.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0019.gif"}" /> and ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0020.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0020.gif"}" /> are ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0021.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0021.gif"}" /> and ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0022.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0022.gif"}" />, then ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0023.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2016/glma20.v064.i11/03081087.2016.1142497/20160822/images/glma_a_1142497_ilm0023.gif"}" /> is a unitary isomorphism or a conjugate unitary isomorphism.     

On closed range for ∂̄

A sufficient condition for ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gcov20/2016/gcov20.v061.i08/17476933.2016.1139579/20170207/images/gcov_a_1139579_ilm0004.gif" alt=" />ipt><img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gcov20/2016/gcov20.v061.i08/17476933.2016.1139579/20170207/images/gcov_a_1139579_ilm0004.gif"}" /> to have closed range is given for pseudoconvex, unbounded domains in ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gcov20/2016/gcov20.v061.i08/17476933.2016.1139579/20170207/images/gcov_a_1139579_ilm0005.gif" alt=" />ipt><img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gcov20/2016/gcov20.v061.i08/17476933.2016.1139579/20170207/images/gcov_a_1139579_ilm0005.gif"}" />.     

Some Topological Properties of Charming Spaces

In this paper, we mainly discuss the class of charming spaces. First, we show that there exists a charming space such that the Tychonoff product is not a charming space. Then we discuss some properties of charming spaces and give some characterizations of some class of charming spaces. Finally, we show that the Suslin number of an arbitrary charming rectifiable space is countable.     

On a Theorem of Dubins and Freedman

Under a notion of <img src="/content/k1v4788x11736750/xxlarge8220.gif" alt="ldquo" align="MIDDLE" BORDER="0">splitting<img src="/content/k1v4788x11736750/xxlarge8221.gif" alt="rdquo" align="MIDDLE" BORDER="0"> the existence of a unique invariant probability, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space <i>Si> generated by iterations of i.i.d. maps on <i>Si>. As corollaries we derive extensions of earlier results of Dubins and Freedman;⁽¹⁷⁾ Yahav;⁽³⁰⁾ and Bhattacharya and Lee⁽⁶⁾ for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins–Freedman result on the <img src="/content/k1v4788x11736750/xxlarge8220.gif" alt="ldquo" align="MIDDLE" BORDER="0">necessity<img src="/content/k1v4788x11736750/xxlarge8221.gif" alt="rdquo" align="MIDDLE" BORDER="0"> of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asymptotic stationarity of the process generated by i.i.d. nondecreasing maps is established without the requirement of continuity. Finally, the theory is applied to the random iteration of two (nonmonotone) quadratic maps each with two repelling fixed points and an attractive period-two orbit.     

The inverse along an element in rings with an involution,Banach algebras and -algebras

Properties of the inverse along an element in rings with an involution, Banach algebras and ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/glma20/2017/glma20.v065.i02/03081087.2016.1183559/20161129/images/glma_a_1183559_ilm0004.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/glma20/2017/glma20.v065.i02/03081087.2016.1183559/20161129/images/glma_a_1183559_ilm0004.gif"}" />-algebras will be studied unifying known expressions concerning generalized inverses.     

Non-trivial 1d and 2d Segal–Bargmann transforms

Special generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as ipt><img src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gitr20/2019/gitr20.v030.i07/10652469.2019.1593407/20190428/images/gitr_a_1593407_ilm0001.gif"}' />ipt><img src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gitr20/2019/gitr20.v030.i07/10652469.2019.1593407/20190428/images/gitr_a_1593407_ilm0001.gif"}" /><img src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />if">i>Li>2-eigenspaces of a special magnetic Schrödinger operator.