Books on the topic 'Bounded linear operators'

This article examines Stefan Banach’s contributions to the field of functional analysis based on the concept of structure and the multiply-flvored expression of generality that arises in his work on linear operations. More specifically, it discusses the two stages in the process by which Banach elaborated a new framework for functional analysis where structures were bound to play an essential role. It considers whether Banach spaces, or complete normed vector spaces, were born in Banach’s first paper, the 1922 doctoral dissertation On operations on abstract spaces and their application to integral equations. It also analyzes what appears to be the core of Banach’s 1922 article and the transformation into a general setting that it represents. The main achievements of Banach’s dissertation, as well as all the essential features that bear witness to the birth of a new theory, are concentrated in the study of linear operations.

In No One's Witness Rachel Zolf activates the last three lines of a poem by Jewish Nazi holocaust survivor Paul Celan—“No one / bears witness for the / witness”—to theorize the poetics and im/possibility of witnessing. Drawing on black studies, continental philosophy, queer theory, experimental poetics, and work by several writers and artists, Zolf asks what it means to witness from the excessive, incalculable position of No One. In a fragmentary and recursive style that enacts the monstrous speech it pursues, No One's Witness demonstrates the necessity of confronting the Nazi holocaust in relation to transatlantic slavery and its afterlives. Thinking along with black feminist theory's notions of entangled swarm, field, plenum, chorus, No One's Witness interrogates the limits and thresholds of witnessing, its dangerous perhaps. No One operates outside the bounds of the sovereign individual, hauntologically informed by the fleshly no-thingness that has been historically ascribed to blackness and that blackness enacts within, apposite to, and beyond the No One. No One bears witness to becomings beyond comprehension, making and unmaking monstrous forms of entangled future anterior life.

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.