Gδ-Embeddings in Hilbert space, II

AbstractIn this paper—which is a continuation of [10]—we exhibit some topological conditions on a Banach space which ensure that it contains isometric copies of infinite-dimensional conjugate spaces. This result is used to identify a large class of Banach spaces that are hereditarily separable duals. A method of defining a “Jamestree sum” of a countable number of Banach spaces is given. It is used to construct various counterexamples; for instance, there exists for each integer n a Banach space that can be mapped into Hilbert space via the composition of n but not (n − 1) Gδ-embeddings. We also continue the investigation of the global structure of some geometrically defined Banach spaces. For example, it is shown that a separable Banach space X with the Radon-Nikodym property (R.N.P.) has a subspace y with a boundedly complete finite-dimensional decomposition (F.D.D.) such that XY has an F.D.D. and the R.N.P

Gδ-embeddings in Hilbert space

AbstractIt is shown that a separable Banach space X has the point of weak to norm continuity property (resp. the Radon-Nikodym property) if and only if there exists a compact Gδ-embedding (resp. an Hδ-embedding) from X into l2. This solves several questions of J. Bourgain and H. P. Rosenthal (J. Funct. Anal. 52 (1983)). It is also shown that every non-relatively compact sequence in a Banach space with property (PC) has a difference subsequence which is a boundedly complete basic sequence. This solves a question of Pelczynski and extends some results of W. B. Johnson and H. P. Rosenthal (Studia Math. 43 (1972), 77–92). Various related questions asked in the above Bourgain-Rosenthal reference and by G. A. Edgar and R. F. Wheeler (Pac. J. Math. 115 (1984)) and N. Ghoussoub and H. P. Rosenthal (Math. Ann. 264 (1983), 321–332) are also settled

Typical entanglement of stabilizer states

How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large the state will be close to maximally entangled with probability exponentially close to one. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few GHZ states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a new concentration inequality which bounds deviations from the mean of random variables which are naturally defined on the Clifford group.Comment: Final version, to appear in PRA. 11 pages, 1 figur

Eigenvalues of p-summing and lp-type operators in Banach spaces

AbstractThis paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ⩾ 2, satisfy ∑n∈N|λn(T)|p1p⩽πp(T). This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π2(n), P = (p1, …, pn) and show that for T ∈ Π2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy ∑i∈N|λi(T)|2nn2⩽πn2(T). More generally, we show that for T ∈ Πp(n), P = (p1, …, pn), pi ⩾ 2, the eigenvalues are absolutely p-summable, 1p=∑i=1n1pi and ∑n∈N|λn(T)|p1p⩽CpπnP(T).We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely ∑n∈N|λn(T)|p ⩽ Cp∑n∈N αn(T)p, 0 < p < ∞, where αn denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space

Coulomb gap in one-dimensional disordered electronic systems

We study a one-dimensional system of spinless electrons in the presence of a long-range Coulomb interaction (LRCI) and a random chemical potential at each site. We first present a Tomonaga-Luttinger liquid (TLL) description of the system. We use the bosonization technique followed by the replica trick to average over the quenched randomness. An expression for the localization length of the system is then obtained using the renormalization group method and also a physical argument. We then find the density of states for different values of the energy; we get different expressions depending on whether the energy is larger than or smaller than the inverse of the localization length. We work in the limit of weak disorder where the localization length is very large; at that length scale, the LRCI has the effect of reducing the interaction parameter K of the TLL to a value much smaller than the noninteracting value of unity.Comment: Revtex, 6 pages, no figures; discussions have been expanded in several place

Interaction effects in multi-subband quantum wires

We investigate the effect of electron-electron interactions on the transport properties of disordered quasi one-dimensional quantum wires with two or more subbands occupied. We apply two alternative methods to solve the logarithmic divergent problem, namely the parquet graph theory and a renormalization group calculation. We solve the group equations analytically in the weak coupling limit and find a power-law for the temperature dependent conductivity of a multi-channel system. The exponent is roughly equal to the inverse of the number of the occupied subbands.Comment: 4 pages, style-files included. No figure. Appears in J. Phys. Soc. Japan (Letter

Coulomb gap in one-dimensional disordered electron systems

The density of states of one-dimensional disordered electron systems with long range Coulomb interaction is studied in the weak pinning limit. The density of states is found to follow a power law with an exponent determined by localization length, and this power law behavior is consistent with the existing numerical results.Comment: RevTeX4 file, 5 pages, no figures To appear in Physical Reviews