In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space M n ( X ) , when X is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of 2 × 2 operator matrices and their off-diagonal parts. One of our main results states that if (X, (On)) is an operator space, then 1 2 max ( W max ( x 1 + x 2 ) , W max ( x 1 x 2 ) ) W max ( 0 x 1 x 2 0 ) 1 2 W max ( x 1 + x 2 ) + W max ( x 1 x 2 ) for all x 1 , x 2 M n ( X ) .

1.
A.
Abu-Omar
and
F.
Kittaneh
, “
Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials
,”
Ann. Funct. Anal.
5
(
1
),
56
62
(
2014
).
2.
M. D.
Choi
,
D. W.
Kribs
, and
K.
Zyczkowski
, “
Quantum error correcting codes from the compression formalism
,”
Rep. Math. Phys.
58
,
77
91
(
2006
).
3.
E. G.
Effros
and
Z.-J.
Ruan
, “
Operator spaces
,” in
London Mathematical Society Monographs
,
New Series
Vol.
23
(
The Clarendon Press, Oxford University Press
,
New York
,
2000
).
4.
K. E.
Gustafson
and
D. K. M.
Rao
, “
Numerical range
,” in
The Field of Values of Linear Operators and Matrices. Universitext
(
Springer-Verlag
,
New York
,
1997
).
5.
O.
Hirzallah
,
F.
Kittaneh
, and
K.
Shebrawi
, “
Numerical radius inequalities for certain 2 × 2 operator matrices
,”
Integral Equations Operator Theory
71
(
1
),
129
147
(
2011
).
6.
O.
Hirzallah
,
F.
Kittaneh
, and
K.
Shebrawi
, “
Numerical radius inequalities for commutators of Hilbert space operators
,”
Numer. Funct. Anal. Optim.
32
,
739
749
(
2011
).
7.
T.
Itoh
and
M.
Nagisa
, “
Numerical radius norms on operator spaces
,”
J. London. Math. Soc.
74
,
154
166
(
2006
).
8.
T.
Itoh
and
M.
Nagisa
, “
Numerical radius Haagerup norm and square factorization through Hilbert spaces
,”
J. Math. Soc. Japan
58
(
2
),
363
377
(
2006
).
9.
F.
Kittaneh
,
M. S.
Moslehian
, and
T.
Yamazaki
, “
Cartesian decomposition and numerical radius inequalities
,”
Linear Algebra Appl.
471
,
46
53
(
2015
).
10.
C. K.
Li
and
Y. T.
Poon
, “
Generalized numerical ranges and quantum error correction
,”
J. Operator Theory.
66
(
2
),
335
351
(
2011
).
11.
C. K.
Li
,
Y. T.
Poon
, and
N.-S.
Sze
, “
Higher rank numerical ranges and low rank perturbations of quantum channels
,”
J. Math. Anal. Appl.
348
(
2
),
843
855
(
2008
).
12.
Z.
Puchała
,
J. A.
Miszczak
,
P.
Gawron
,
C. F.
Dunkl
,
J. A.
Holbrook
, and
K.
Zyczkowski
, “
Restricted numerical shadow and the geometry of quantum entanglement
,”
J. Phys. A
45
(
41
),
415309
(
2012
).
13.
Z-J.
Ruan
, “
subspaces of C-algebras
,”
J. Funct. Anal.
76
,
217
230
(
1988
).
14.
M.
Sattari
,
M. S.
Moslehian
, and
T.
Yamazaki
, “
Some generalized numerical radius inequalities for Hilbert space operators
,”
Linear Algebra Appl.
470
(
1
),
216
227
(
2015
).
You do not currently have access to this content.