Let A be a positive bounded operator on a Hilbert space H, ·, · . The semi-inner product x, y A := Ax, y , x, y ∈ H induces a semi-norm · A on H. Let T A and w A (T ) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space H, · A , respectively. In this paper, we prove the following characterization of w A (T ) 2010 Mathematics Subject Classification. Primary 47A05; Secondary 46C05, 47B65, 47A12.

Combining edge processing (at data capture site) with analysis carried out while data is enroute from the capture site to a data center offers a variety of different processing models. Such in-transit nodes include network data centers that have generally been used to support content distribution (providing support for data multicast and caching), but have recently started to offer user-defined programmability, through Software Defined Networks (SDN) capability, e.g. OpenFlow and Network Function Visualization (NFV). We demonstrate how this multi-site computational capability can be aggregated to support video analytics, with Quality of Service and cost constraints (e.g. latency-bound analysis). The use of SDN technology enables separation of the data path from the control path, enabling in-network processing capabilities to be supported as data is migrated across the network. We propose to leverage SDN capability to gain control over the data transport service with the purpose of dynamically establishing data routes such that we can opportunistically exploit the latent computational capabilities located along the network path. Using a number of scenarios, we demonstrate the benefits and limitations of this approach for video analysis, comparing this with the baseline scenario of undertaking all such analysis at a data center located at the core of the infrastructure.

Dramatic changes in the technology landscape marked by increasing scales and pervasiveness of compute and data have resulted in the proliferation of edge applications aimed at effectively processing data in a timely manner. As the levels and fidelity of instrumentation increases and the types and volumes of available data grow, new classes of applications are being explored that seamlessly combine real-time data with complex models and data analytics to monitor and manage systems of interest. However, these applications require a fluid integration of resources at the edge, the core, and along the data path to support dynamic and data-driven application workflows, that is, they need to leverage a computing continuum. In this article, we present our vision for enabling such a computing continuum and specifically focus on enabling edge-to-cloud integration to support data-driven workflows. The research is driven by an online data-driven tsunami warning use case that is supported by the deployment of large-scale national environment observation systems. This article presents our overall approach as well as current status and next steps.

Let A be a positive bounded operator on a Hilbert space H, ·, · . The semi-inner product x, y A := Ax, y , x, y ∈ H, induces a seminorm · A on H. Let T A , w A (T ), and c A (T ) denote the A-operator seminorm, the A-numerical radius, and the A-Crawford number of an operator T in the semi-Hilbertian space H, · A , respectively. In this paper, we present some seminorm inequalities and equalities for semi-Hilbertian space operators. More precisely, we give some necessary and sufficient conditions for two orthogonal semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive new upper and lower bounds for the numerical radius of operators in semi-Hilbertian spaces. In particular, we show that

Abstract. Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(H ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on B(H ). We also characterize the parallel elements of a C * -algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert C * -module.