NORM INEQUALITIES FOR ACCRETIVE-DISSIPATIVE BLOCK MATRICES

Let T = [Tij ] \in \BbbM mn(\BbbC ) be accretive-dissipative, where Tij \in \BbbM n(\BbbC ) for i, j = 1, 2, ...,m. Let f be a function that is convex and increasing on [0,\infty ) where f(0) = 0. Then \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \left(  \sum i<j | Tij | \right) + f \left( \sum i<j \bigm| \bigm| T \ast ji\bigm| \bigm| 2 \right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq  \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \biggl( m2  m 2 | T | \biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . Also, if f is concave and increasing on [0,\infty ) where f(0) = 0, then \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \left(  \sum i<j | Tij | \right) + f \left( \sum i<j \bigm| \bigm| T \ast ji\bigm| \bigm| 2 \right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq (2m2  2m) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \Biggl( | T | 4 \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . ¥å © T = Tij \in \BbbM mn(\BbbC ), ¤¥ Tij \in \BbbM n(\BbbC )  ̄à ̈ i, j = 1, 2, ...,m., {  aà¥â ̈¢­®¤ ̈á ̈ ̄ â ̈¢­  ¬ âà ̈æï. ¥å © f ® ̄ãa«  äã­aæ÷ï, ïa  §à®áâ õ ­  [0,\infty ), ¤¥ f(0) = 0. ®¤÷ \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \left(  \sum i<j | Tij | \right) + f \left( \sum i<j \bigm| \bigm| T \ast ji\bigm| \bigm| 2 \right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq  \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \biggl( m2  m 2 | T | \biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .  a®¦, ïaé® f õ ã£­ãâ®î, §à®áâ õ ­  [0,\infty ) ÷ f(0) = 0, â® \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \left(  \sum i<j | Tij | \right) + f \left( \sum i<j \bigm| \bigm| T \ast ji\bigm| \bigm| 2 \right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq (2m2  2m) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \Biggl( | T | 4 \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .

A principal submatrix of a square matrix A is the matrix obtained by deleting any j rows and the corresponding j columns.

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In this paper, some norm inequalities concerning with accretive-dissipative block matrices in \BbbM mn (\BbbC ) (m \in \{ 2, 3, 4, ...\} ) are given. In Section 2, some unitarily invariant norm inequalities that compare the accretive-dissipative matrix T to its o-diagonal blocks, where T is partitioned as in (1.1) are derived. In Section 3, a unitarily invariant norm inequalities for functions f \in \zeta are presented. In Section 4, some results for a 2 \times 2 accretive-dissipative block matrices are given.

Some unitarily invariant norm inequalities
In this section, we give some unitarily invariant norm inequalities that compare the accretive-dissipative matrix T to its o-diagonal blocks, where T is partitioned as in (1.1). To start our work, we will use the following lemma (see [13]).
In the following lemma, part (a) is an extension of Theorem 2.3 in [1] for n-tuples of operators (see also [9, Theorem 1]), a stronger version of part (b) of the lemma can be obtained by invoking an argument similar to that used in the proof of Proposition 4.1 in [14]. For various Jensen type matrix inequalities, we refer to [3] and references therein. Part (c) can be found in [11] and we can nd part (d) in [4]. Henceforth, we assume that every function is continuous.   for every function f that is concave and nonnegative on [0, \infty ).
Our rst main result in this section is the following theorem. (2.1) . (2.8) In particular, when m = 2, we have Proof. The inequality ( Proof. The proof follows by applying the inequality (2.1) to the function f (t) = e t -1 which is a convex function that is increasing on [0, \infty ) with f (0) = 0.

\square
The following corollary can be obtained by applying Theorems 2.6 and 2.10 to the function f (t) = t p/2 . Corollary 2.13. Let T \in \BbbM mn (\BbbC ) be a partitioned accretive-dissipative matrix as given in (1.1

Unitarily invariant norm inequalities involving a special class of functions
In this section, we give unitarily invariant norm inequalities including functions belongs to the class \zeta .
We start this section with the following lemma (see [8]).
\in \BbbM 2n (\BbbC ) be positive semidenite, and let f \in \zeta be submultiplicative function. If p and q are positive real numbers with 1 Our rst result in this section is the following theorem.
\square Note that the inequality (1.5) follows by taking m = 2 in the inequality (3.2). So, the inequality (3.2) gives a generalization to the inequality (1.5). Theorem 3.3. Let T \in \BbbM mn (\BbbC ) be a partitioned accretive-dissipative matrix as given in (1.1), and let f \in \zeta be a function that is concave and submultiplicative and satisfying that f (0) = 0. If p, q > 0 satisfying 1 p +