We study the topic of quantum differentiability on quantum Euclidean
d
-dimensional spaces (otherwise known as Moyal
d
-spaces), and we find conditions that are necessary and sufficient for the ...singular values of the quantised differential
to have decay
O
(
n
-
α
)
for
0
<
α
≤
1
d
. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum
d
-tori.
We study the class of functions
f
on
R
satisfying a Lipschitz estimate in the Schatten ideal
L
p
for
0
<
p
≤
1
. The corresponding problem with
p
≥
1
has been extensively studied, but the ...quasi-Banach range
0
<
p
<
1
is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class
B
˙
p
1
-
p
,
p
1
p
(
R
)
obey the estimate
‖
f
(
A
)
-
f
(
B
)
‖
p
≤
C
p
(
‖
f
′
‖
L
∞
(
R
)
+
‖
f
‖
B
˙
p
1
-
p
,
p
1
p
(
R
)
)
‖
A
-
B
‖
p
for all bounded self-adjoint operators
A
and
B
with
A
-
B
∈
L
p
. In the case
p
=
1
, our methods recover and provide a new perspective on a result of Peller that
f
∈
B
˙
∞
,
1
1
is sufficient for a function to be Lipschitz in
L
1
. We also provide related Hölder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on
R
are not Lipschitz in
L
p
for any
0
<
p
<
1
. This gives counterexamples to a 1991 conjecture of Peller that
f
∈
B
˙
∞
,
p
1
/
p
(
R
)
is sufficient for
f
to be Lipschitz in
L
p
.
Let E be a symmetric Banach sequence space. We show that there exists an equivalent symmetric norm on E which is monotone with respect to the Weyl (i.e., logarithmic) submajorization. Surprisingly, ...this purely commutative result is proved by a very non-commutative method.
Optimal Cwikel–Solomyak Estimates Sukochev, Fedor; Zanin, Dmitriy
The Journal of fourier analysis and applications,
04/2023, Volume:
29, Issue:
2
Journal Article
Peer reviewed
Open access
We obtain the optimal version of Cwikel-type estimates for the uniform operator norm which implies the optimality of M.Z. Solomyak’s results (Proc. Lond. Math. Soc. 71(1):53–75, 1995) within the ...classes of Orlicz/Lorentz spaces. Our methods are based on finding the distributional version of the Sobolev inequality.
Asymptotics of singular values for quantum derivatives Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy
Transactions of the American Mathematical Society,
March 1, 2023, 2023-3-00, Volume:
376, Issue:
3
Journal Article
Peer reviewed
Open access
We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f from the homgeneous Sobolev space \dot {W}^1_d(\mathbb {R}^d) on \mathbb {R}^d. The asymptotic coefficient ...\|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C^{\ast }-algebraic notion of the principal symbol mapping on \mathbb {R}^d, as developed recently by the last two authors and collaborators.
We resolve an open problem due to B. Simon 7 concerning certain Cwikel-type estimates for Schrödinger operators. We provide a negative resolution already for the case of Laplacian.
We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a 2-co-lacunary subsequence. This extends the classical ...characterization, due to Räbiger.
Let
M
be a semifinite von Neumann algebra. We show that an operator
T
from the predual
L
1
(
M
,
τ
)
of
M
into a Banach space
X
is strongly Dunford–Pettis if and only if
T
∘
i
:
L
1
(
M
,
τ
)
∩
M
→
i
...L
1
(
M
,
τ
)
→
T
X
is compact. We also show that for a finite measure space
(
A
,
Σ
,
ν
)
and a reflexive Banach space
X
, a linear bounded operator
T
:
L
1
(
ν
,
X
)
→
c
0
is a Dunford–Pettis operator if and only if
T
is a dominated operator. We also prove that all bounded operators
T
from
L
p
(
0
,
1
)
into the Schatten–von Neumann
r
-class
C
r
are necessarily narrow whenever
1
≤
p
<
2
,
1
≤
r
≤
p
, which answers a question raised in Huang et al. (Mediterr. J Math 19, 2022). Finally, we show that for a reflexive Banach space
X
all Dunford–Pettis operators from
T
:
L
1
(
ν
,
X
)
→
c
0
are narrow.
Consider the generalized absolute value function defined by
a
(
t
)
=
|
t
|
t
n
−
1
,
t
∈
ℝ
,
n
∈
ℕ
≥
1
.
.
Further, consider the
n
-th order divided difference function
a
n
: ℝ
n
+1
→ ℂ and let 1 ...<
p
1
, …,
p
n
< ∞ be such that
∑
l
=
1
n
p
l
−
1
=
1
. Let
S
p
l
denote the Schatten-von Neumann ideals and let
S
1
,
∞
denote the weak trace class ideal. We show that for any (
n
+ 1)-tuple
A
of bounded self-adjoint operators the multiple operator integral
T
a
n
A
maps
S
p
1
×
⋯
×
S
p
n
to
S
1
,
∞
boundedly with uniform bound in
A
. The same is true for the class of
C
n
+1
-functions that outside the interval −1, 1 equal
a
. In CLPST16 it was proved that for a function {at
f
} in this class such boundedness of
T
f
n
A
from
S
p
1
×
⋯
×
S
p
n
to
S
1
may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences.
Let \mathcal {H} be a separable Hilbert space and let B(\mathcal {H}) be the *-algebra of all bounded linear operators on \mathcal {H}. In the present paper, we prove that a positive/regular operator ...from L_1(0,1) into an arbitrary separable operator ideal in B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy Glasgow Math. J. 28 (1986), pp. 113–114, and Holub Proc. Amer. Math. Soc. 104 (1988), pp. 89–95. Consequently, for an arbitrary atomless von Neumann algebra \mathcal {M} and an arbitrary KB -ideal C_E in B(\mathcal {H}), the predual \mathcal {M}_* of \mathcal {M} is not isomorphic to any subspace of C_E. This observation complements several earlier results.
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