In this paper, we introduce and study the mixed central bounded mean oscillation space , which is a new central version of the bounded mean oscillation spaces BMO. The pre-dual of is proved to be the ...mixed Herz–Hardy space . Moreover, we also give a characterization of via the boundedness of the commutators of n-dimensional Hardy operator and its dual operator on mixed Herz spaces .
Finite groups in which every element has prime power order (EPPO-groups) are nowadays fairly well understood. For instance, if G is a soluble EPPO-group, then the Fitting height of G is at most 3 and ...|π(G)|⩽2 (Higman, 1957). Moreover, Suzuki showed that if G is insoluble, then the soluble radical of G is a 2-group and there are exactly eight nonabelian simple EPPO-groups.
In the present work we concentrate on finite groups in which every commutator has prime power order (CPPO-groups). Roughly, we show that if G is a CPPO-group, then the structure of G′ is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPO-group is at most 3 and |π(G′)|⩽3. Moreover, if G is insoluble, then R(G′) is a 2-group and G′/R(G′) is isomorphic to a simple EPPO-group.
Commutators on ℓ 1 Dosev, Detelin T.
Journal of functional analysis,
06/2009, Letnik:
256, Številka:
11
Journal Article
Recenzirano
Odprti dostop
The main result is that the commutators on
ℓ
1
are the operators not of the form
λ
I
+
K
with
λ
≠
0
and
K compact. We generalize Apostol's technique C. Apostol, Rev. Roumaine Math. Appl. 17 (1972) ...1513–1534 to obtain this result and use this generalization to obtain partial results about the commutators on spaces
X
which can be represented as
X
≃
(
⊕
i
=
0
∞
X
)
p
for some
1
⩽
p
<
∞
or
p
=
0
. In particular, it is shown that every compact operator on
L
1
is a commutator. A characterization of the commutators on
ℓ
p
1
⊕
ℓ
p
2
⊕
⋯
⊕
ℓ
p
n
is given. We also show that strictly singular operators on
ℓ
∞
are commutators.
Counterdiabatic (CD) driving presents a way of generating adiabatic dynamics at an arbitrary pace, where excitations due to nonadiabaticity are exactly compensated by adding an auxiliary driving term ...to the Hamiltonian. While this CD term is theoretically known and given by the adiabatic gauge potential, obtaining and implementing this potential in many-body systems is a formidable task, requiring knowledge of the spectral properties of the instantaneous Hamiltonians and control of highly nonlocal multibody interactions. We show how an approximate gauge potential can be systematically built up as a series of nested commutators, remaining well defined in the thermodynamic limit. Furthermore, the resulting CD driving protocols can be realized up to arbitrary order without leaving the available control space using tools from periodically driven (Floquet) systems. This is illustrated on few- and many-body quantum systems, where the resulting Floquet protocols significantly suppress dissipation and provide a drastic increase in fidelity.
In this paper, we introduce and study maximal submultigroups and present some of its algebraic properties. Frattini submultigroups as an extension of Frattini subgroups is considered. A few ...submultigroups results on the new concepts in connection to normal, characteristic, commutator, abelian and center of a multigroup are established and the ideas of generating sets, fully and non-fully Frattini multigroups are presented with some significant results.
The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of ...their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.
In recent years, it has been well understood that a Calderón–Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse ...operators). We obtain a similar pointwise estimate for the commutator b,T with a locally integrable function b. This result is applied into two directions. If b∈BMO, we improve several weighted weak type bounds for b,T. If b belongs to the weighted BMO, we obtain a quantitative form of the two-weighted bound for b,T due to Bloom–Holmes–Lacey–Wick.
A remark on a paper of E. B. Davies Zworski, Maciej
Proceedings of the American Mathematical Society,
10/2001, Letnik:
129, Številka:
10
Journal Article
Recenzirano
Odprti dostop
We explain the existence of open sets of complex quasi-modes in terms of Hörmander's commutator condition.
The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling ...of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure,k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
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