
\documentclass[10pt,leqno,twoside]{siamltex}
\usepackage{amsfonts}
\setlength{\textwidth}{130mm}
\setlength{\textheight}{190mm}
\setlength{\topmargin}{20mm}

\begin{document}
\input{elaheader-vol13-8.tex}
\setcounter{page}{122}

\bibliographystyle{plain}

%\setlength{\leftmargin}{40mm}
%\newcommand{\bc}{\mathbb{C}}

\newcommand{\n}[1]{\|#1 \|}
\newcommand{\eq}{\leqno(}

\newcommand{\br}{\mathbb{R}}
\newcommand{\bc}{\mathbb{C}}
\newcommand{\rn}{\mathbb{R}^n}
\newcommand{\cn}{\mathbb{C}^n}

\newcommand{\ot}{\otimes}

\newcommand{\fd}[1]{\frac{d}{d #1 }}
\newcommand{\fdf}[2]{\frac{d #1}{d #2}}
\newcommand{\fdmf}[3]{\frac{d^{#1} #2}{d #3^{#1}}}
\newcommand{\fdm}[2]{\frac{d^{#1}}{d #2^{#1} }}

\newcommand{\pa}{\partial}
\newcommand{\fp}[1]{\frac{\partial}{\partial #1 }}
\newcommand{\fpm}[2]{\frac{\partial^#1 }{\partial #2^#1}}
\newcommand{\fpf}[2]{\frac{\partial #1 }{\partial #2}}
\newcommand{\fpmf}[3]{\frac{\partial^#1 #2 }{\partial #3^#1}}

\newcommand{\bs}[2]{\sum_{k=#1}^{#2}}
\newcommand{\bsj}[2]{\sum_{j=#1}^{#2}}
\newcommand{\bst}[2]{\sum_{t=#1}^{#2}}
\newcommand{\bsd}[4]{\sum_{j=#1}^{#2}\sum_{k=#3}^{#4}}


\newcommand{\bi}[2]{\int_{#1}^{#2}}
\newcommand{\bsu}[2]{\sup_{#1\in #2}}
\newcommand{\bin}[2]{\inf_{#1\in #2}}
\newcommand{\bli}[2]{\lim_{#1\rightarrow #2}}
\newcommand{\bp}[2]{\prod_{k=#1}^{#2}}
\newcommand{\bma}[1]{\max_{k=1, ..., #1}}
\newcommand{\bmi}[1]{\min_{k=1, ..., #1}}


\newcommand{\ti}{\tilde}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\ovl}[1]{\overline{#1}}
\newcommand{\8}{\infty}
\newcommand{\Rar}{\Rightarrow}
\newcommand{\nRaro}{\hbox{$\Rightarrow$}}
\newcommand{\Lar}{\Leftrightarrow}
\newcommand{\rar}{\rightarrow}
\newcommand{\lar}{\leftarrow}
\newcommand{\dar}{\downarrow}

\newcommand{\mi}{^{-1}}
\newcommand{\fmi}[1]{\frac{1}{#1}}
\newcommand{\fmp}{\frac{1}{2\pi i}}


\newcommand{\dline}[1]{\| #1 \|}
\newcommand{\ab}{\hspace*{5 mm}}
\newcommand{\ac}{\vspace*{3 mm}}

\newcommand{\nin}{\not\in}
\newcommand{\nos}{\not\in\si(A)}
\newcommand{\<}{\subset}
\newcommand{\sps}{\supset}

\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\vep}{\varepsilon}
\newcommand{\ze}{\zeta}
\newcommand{\et}{\eta}
\newcommand{\te}{\theta}
\newcommand{\Th}{\Theta}
\newcommand{\vth}{\vartheta}
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\ka}{\kappa}
\newcommand{\vpi}{\varpi}
\newcommand{\rh}{\rho}
\newcommand{\vrh}{\varrho}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\vsi}{\varsigma}
\newcommand{\ta}{\tau}
\newcommand{\up}{\upsilon}
\newcommand{\Up}{\Upsilon}
\newcommand{\ph}{\phi}
\newcommand{\Ph}{\Phi}
\newcommand{\vph}{\varphi}
\newcommand{\ps}{\psi}
\newcommand{\Ps}{\Psi}
\newcommand{\ch}{\chi}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}

\newcommand{\Bo}{$\Box$\newline}
%\newcommand{\Bo}{ Q.E.D.\newline}


%\newcommand{\proof}{{\bf Proof: }}
%\newcommand{\proof}{\par\noindent{\bf Proof:}\quad}
\newcommand{\proofo}{\par\noindent{\bf Proof}\quad}
\newcommand{\remarks}{\par\noindent{\bf Remark:}\quad}


\newtheorem{cor}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\title{NORM ESTIMATES FOR FUNCTIONS OF TWO COMMUTING MATRICES\thanks{Received by the 
editors 26 September 2004. Accepted for publication 6 March 2005.   
Handling Editor: Harm Bart.}}

\author{Michael Gil'\thanks{Department of Mathematics,
Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
(gilmi@cs.bgu.ac.il).}}

\pagestyle{myheadings} \markboth{M.\ Gil'}{Norm Estimates}

\maketitle

\begin{abstract}
Matrix valued analytic functions of two commuting matrices are 
considered. A precise  norm estimate is established.  
As a particular case, 
the matrix valued functions of two  matrices on  tensor products of  
Euclidean spaces are explored. 
\end{abstract}

\begin{keywords}   
Functions of commuting matrices, Norm estimate.
\end{keywords}

\begin{AMS} 
15A15, 15A54, 15A69, 15A60.
\end{AMS}

%-------------------------------------------------------------

\section{Introduction and statement of the main result}

In the book \cite{ge}, I.M. Gel'fand and G.E. Shilov 
have established
an estimate for the norm of a regular matrix-valued
function   in connection
with their investigations of partial differential equations. However  that
estimate is not sharp, it is not attained for any matrix. The problem of
obtaining a precise estimate for the norm of a matrix-valued function has been
 repeatedly discussed in the literature. In the paper \cite{gi93} (see  also \cite{gi03}) 
the author has derived a precise estimate for a regular matrix-valued function. It is
attained in the case of normal matrices. In the present paper we generalize
the main result of the paper \cite{gi93} to functions of two commuting matrices.  
Besides, the main result of the present paper-Theorem 1.1 is improved 
in the case of   matrices on  tensor products of  Euclidean spaces.


It should be noted that functions of commuting operators were investigated
by many mathematicians, cf. \cite{ar,mu,ta} and references therein however
the norm estimates were not considered, but 
as it is well-known, matrix valued functions are 
Green's functions and characteristic functions of various differential and difference 
equations. This fact allow us to 
investigate stability, well-posedness and perturbations of these equations
by  norm estimates for matrix valued functions, cf. \cite{da,ges,gi98}. 


Let  $\cn$ be a Euclidean space  with a scalar product $(\cdot,\cdot)$, the unit matrix $I$
and the Euclidean norm
$\|\cdot\|=(\cdot,\cdot)^{1/2}$; $ M(\cn)$ denotes the set of all linear operators in $\cn$.
For a $A\in M(\cn)$, $\|A\|$ is the operator norm;
 $N(A)$ is the Frobenius (Hilbert-Schmidt) norm: $ N^2(A)=Trace\; (AA^{*})$;
$A^*$ is the operator adjoint to $A$,
$\la_j(A),\; j=1,\dots,n $ are the eigenvalues counting with their multiplicities, $\si(A)$ 
is the spectrum.


Everywhere  below $A$ and $B$ are commuting matrices.
 Let $\Om_A$ and $\Om_B$ 
be open simple connected sets containing  $\si(A)$ and $\si(B)$, respectively.
Let $f$ be a scalar function analytic on $\Om_A\times\Om_B$. 
Introduce the operator valued function
$$
f(A, B):=-\frac{1}{4\pi^2}\int_{L_B} \int_{L_A} f(z, w) R_{z}(A)  R_{w}(B)
dw\;dz,
\eq 1.1)
$$
where  $L_A\subset \Om_A, L_B\subset \Om_B$ 
are closed contour surrounding $\si(A)$ and $\si(B)$, respectively.
Note that if the series 
$$
f(z, w)=\sum_{j, k=0}^\8 c_{jk} z^{j}w^k
$$
converges for $|z|\leq r_s(A), |w|\leq r_s(w)$, where
$r_s(A)$ denotes the spectral radius of $A$, 
then (1.1) holds.




The following quantity plays a key role in the sequel
    $$
    g(A)=(N^2(A)-\sum^n_{k=1}|\la_k(A)|^2)^{1/2}.
    $$
Since
$$
\sum^n_{k=1}|\la_k(A)|^2\geq |Trace\; A^2|,
\mbox{ we have }g^2(A)\leq N^2(A)-|Trace\; A^2|.
$$
 If $A$ is a normal matrix, i.e. if $ AA^*=A^*A $, then $ g(A)=0 $. 
 Also the inequality
    $$
    g^2(A)\leq \frac{1}{2} N^2(A^*-A)
    %\eqno(1.2)
    \eq 1.2)
    $$
   is valid, cf.  \cite[Section 2.1]{gi03}.
Introduce the numbers
$$
\et_k:= \frac{1}{k!}\sqrt{\frac{(n-1)!}{(n-k-1)!(n-1)^k}}\mbox{    for  }k=1,..., n-1;\;
\et_{0}=1.
$$
It is simple to check that
$$
\et^2_{k}\leq \frac{1}{(k!)^{3}}\;\;(k=1,..., n-1).
\eq 1.3)
$$    
Denote by 
$co(A), co(B)$ the closed convex hulls of $\si(A)$ and $\si(B)$, respectively.
Put
$$
f^{(j, k)}(z, w)=\frac{\pa^{j+k}f(z,w)}{\pa z^{j}\pa w^{k}}.
$$
Now we are in a position to formulate the main result of the paper.
\begin{theorem} Let $A$ and $B$ be commuting $n\times n$-matrices 
and $f(z,w)$ be regular on a neighborhood of $co(A)\times co(B)$. Then
 $$
 \|f(A, B)\|\leq \sum_{ j, k= 0}^{j+k \leq  n-1}
\et_j\et_k g^j(A) g^k(B)
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|.
    $$
 \end{theorem}
The proof of this theorem is divided into a series of lemmas which are 
presented in the next  section. If both $A$ and $B$ are normal operators, and
$$
\sup_{z\in co(A), w\in co(B)}|f(z, w)|=\sup_{z\in \si(A), w\in \si(B)}|f(z, w)|,
$$
then Theorem 1.1 gives us the exact relation
$$
 \|f(A, B)\|=\sup_{z\in \si(A), w\in \si(B)}|f(z, w)|.
$$
Taking into account (1.3), we get
\begin{cor} Under the hypothesis of {\rm Theorem 1.1},
the estimate
$$
 \|f(A, B)\|\leq \sum_{j,k= 0}^{j+k\leq n-1}
\frac{g^j(A) g^k(B)}{(j!k!)^{3/2}}
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|
$$
is true.
    \end{cor}

Let $A$ be a normal matrix. Then $g(A)=0$. Now Corollary  1.2 implies
$$
\|f(A, B)\|\leq \sum_{0\leq k\leq n-1}\frac{g^k(B)}{(k!)^{3/2}}
\sup_{z\in co(A), w\in co(B)}|\frac{\pa^{k}f(z,w)}{\pa w^{k}}|.
$$

Let us evaluate the error of Theorem 1.1 in the case of non-normal matrices. 
Certainly, we can  obtain the exact value of the norm 
of a function of two matrices  in  very simple cases only.
Consider the $2\times 2$-matrices
$$
A= \left(\begin{array}{cc}
a &1/3\\
0 &a
\end{array}
\right )
.
$$
and $B=2A$ with $a>0$. Construct a function of two variables by 
setting $(x, y)\to f(x+y)$. Direct calculations show that
$$
f(A+B)= \left(\begin{array}{cc}
f(3a) &f'(3a)\\
0 &f(3a)
\end{array}
\right )
.
$$
Assume that $f(3a)=0$. Then $\|f(A+B)\|=|f'(3a)|$.
At the same time, Theorem 1.1 gives us the relations,
$$
\|f(A+B)\|\leq |f(a)|+|f'(3a)|(g(A)+g(B))=|f'(3a)|,
$$
since $g(A)=1/3,\;g(B)=2/3$. Thus in the considered case Theorem 1.1 gives us
the exact result. 
\begin{example}
{\rm 
Consider the polynomial} 
\end{example}
$$
P(z,w)=\sum_{\nu=0}^{m_1}\sum_{l=0}^{m_2}c_{\nu l}z^\nu w^l
$$ 
with complex, in general,   coefficients. Then
$$
P^{(j, k)}(z, w)=
\sum_{\nu=0}^{m_1-j}\sum_{l=0}^{m_2-k} c_{\nu l} \nu(\nu-1)... (\nu-j+1)z^{\nu-j}l... (l-k+1)w^{l-k}.
$$
Now Theorem 1.1 implies
 $$
\|P(A, B)\|\leq \sum_{j,k= 0}^{j+k \leq n-1}
\et_j\et_k g^j(A) g^k(B)\sum_{\nu=0}^{m_1-j}\sum_{l=0}^{m_2-k}
\frac{\nu! l!|c_{\nu l}|}{(l-k)!(\nu-j)!}
r_s^{\nu-j}(A)r_s^{l-k}(B).
$$
Recall that $r_s(.,)$ denotes the spectral radius.
 If both $A$ and $B$ are normal operators,
then
 $$
\|P(A, B)\|\leq \sum_{\nu=0}^{m_1}\sum_{l=0}^{m_2}|c_{\nu l}|r_s^{\nu}(A)r_s^{l}(B).
$$

\begin{example}{\rm 
Consider  the function}
\end{example}
$$
f(z, w)=cos\; (xz+ yw )\;\;(y, x\geq 0).
$$
Note that the function $U(x,y)=cos (xA+ yB )$ is a solution of the equation
$$
\pa^2 U(x,y)/\pa x^2 + \pa^2 U(x,y)/\pa y^2 + (A^2+B^2)U(x,y)=0. 
$$
In the considered case
$$
|f^{(j, k)}(z, w)|=x^{j}y^{k}|cos\; (xz+ yw )|\;\;(j+k\mbox{ is even }),
$$
$$
|f^{(j, k)}(z, w)|=x^{j}y^{k}|sin\; (xz+ yw )|\;\;(j+k\mbox{ is odd }). 
$$
For simplicity assume that  the spectra of both $A$ and $B$ are real. Then 
thanks to Theorem 1.1,
$$
\|U(x, y)\|\leq \sum_{j, k=0}^{j+k \leq  n-1}
\et_j\et_k g^j(A) g^k(B)x^jy^k\;\;(x, y\geq 0).
$$

\section{Proof of Theorem 1.1}
The following lemma is needed.

\begin{lemma} Let  $\Om $ and $\ti \Om$ be the closed convex hulls  
of complex, in general, points 
$$
x_0, x_1, ... , x_{n}
\eq 2.1)
$$
and
$$
y_0, y_1, \dots  , y_{m},
\eq 2.2)
$$ 
respectively, and let a scalar-valued function $f(z, w)$ be regular on  
 $D\times \ti D$, where $D$ and $\ti D$ are neighborhoods of
$\Om $ and $\ti \Om$, respectively. 
In addition, let  $L\subset D, \ti L\subset \ti D$ be  Jordan closed contours 
 surrounding the points in {\rm (2.1)} and {\rm (2.2)}, respectively. 
Then with the notation  
$$
Y(x_0,...,  x_{n};\;y_0,... y_{m})=
-\frac{1}{4\pi^2} \int_{L} \int_{\ti L} \frac{f(z, w) dz\;dw}
{(z-x_0)\cdots  (z-x_{n})(w-y_0)\cdots  (w-y_{m})},
$$
we have
$$
|Y(x_0,...,  x_{n};\;y_0,... y_{m})|
\leq 
\frac{1}{n !m!} 
\;\sup_{z\in \Om, w\in \ti \Om} |f^{(n, m)}(z, w)|.
$$
\end{lemma}

\begin{proof}
First, let all the points be distinct: $x_j\neq x_k,\;y_j\neq y_k$ for 
$j\neq k$.  Let
a  function $h$ of one variable be regular on $D$
and  $[x_0, x_1, ... , x_n ]h$ be a divided difference of 
function $h$ at points $x_0, x_1, ... , x_n$.
Then 
$$
[x_0, x_1, ... , x_n ]h=
\fmp \int_{L} \frac{h(\la) d\la}{(\la-x_0)\cdots   
(\la-x_n)}
\eq 2.3)
$$
(see \cite[ formula (54)]{ge}).
Thus
$$
Y(x_0,... x_{n};\;y_0,..., y_m)=
\frac{1}{2\pi i} \int_{\ti L} \frac{[x_0,..., x_{n}]f(.,w)dw} 
{(w-y_0)\cdots   (w-y_n)}.
$$
Now apply (2.3) to $[x_0,... x_{n}]f(.,w)$. Then
$$
Y(x_0,... x_{n};\;y_0,..., y_m)=
[x_0,... x_{n}]\;[y_0,... y_m]f(.,.)\equiv
[x_0,... x_{n}]\;([y_0,... y_m]f(.,.)). 
$$
The following estimate is well-known \cite[ p. 6 ]{os}:
$$
|[x_0,... x_{n}]\;[y_0,... y_m]f(.,.)|
\leq\;\sup_{z\in \Om, w\in \ti \Om} |f^{(n, m)}(z, w)|.
$$
It proves the required result if all the points are distinct.
 If some points coincide,
then the claimed inequality can be obtained by small perturbations and the 
previous arguments. 
\end{proof}





Furthermore,  since $A$ and $B$ commute
they have the same orthogonal normal basis of the triangular 
representation (Schur's basis)  $\{e_k\}$. We can write
$$
A=D_A+V_A, 
B=D_B+V_B,
\eq 2.4)
$$
where $D_A, D_B$ are the diagonal parts, $V_A$ and $V_B$
are the nilpotent parts of $A$ and $B$, respectively.
Furthermore, let $|V_A|$ be the operator whose matrix elements in $\{e_k\}$
 are the absolute values of the matrix elements of the nilpotent
part $V_A$ of $A$ with respect to this basis. That is,
 $$
 |V_A|=\sum_{k=1}^{n} \sum_{j=1}^{k-1}|a_{jk}|(.,e_k)e_j,
 $$
    where  $a_{jk}=(Ae_k, e_j)$.
Similarly $|V_B|$ is defined.

    \begin{lemma} Under the hypothesis of {\em Theorem 1.1}
   the estimate
    $$
    \|f(A, B)\|\leq \sum_{j,k= 0}^{j+k \leq  n-1}
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|
\frac{\|\;|V_A|^j \;|V_B|^k\|}{j!k!}
    $$
    is true, where $V_A$ and $V_B$ are  the nilpotent part of $A$ and $B$, respectively.
    \end{lemma}

\begin{proof}  
It is not hard to see that
    the representation (2.4) implies the equality
    $$
    (A-I \la)\mi=(D_A+V_A-\la I)\mi =(I+R_{\la}(D_A)V_A)\mi
    R_{\la}(D_A)
    $$
for all regular $\la$. According to Lemma 1.7.1 from \cite{gi03} $R_{\la}(D_A)V_A$ is a
nilpotent operator because $V$
    and $R_{\la}(D_A)$ the same invariant subspaces. Hence, $(R_{\la}(D_A)V_A)^{n}=0$.
    Therefore,
    $$
 R_{z}(A)=\sum_{k=0}^{n-1}(R_z(D_A)V_A)^k(-1)^k R_{z}(D_A).
    $$
Similarly,
    $$
 R_{\mu}(B)=\sum_{k=0}^{n-1}(R_\mu(D_B)V_B)^k(-1)^k R_{\mu}(D_B).
    $$
So from (1.1) it follows
$$
f(A, B)=\sum_{j,k=0}^{n-1}C_{jk},
%\eqno (2.5)
\eq 2.5)
$$
where
$$
C_{jk}=\frac{(-1)^{k+j} }{4\pi^2}
\int_{L_B} \int_{L_A} f(z, w)(R_{z}(D_A)V_A)^j R_{z}(D_A) 
(R_{w}(D_B)V_B)^k R_{w}(D_B)dz\; dw. 
$$ 
 Since $D_A$ is a diagonal matrix with respect to the Schur basis $\{e_k\}$
and its diagonal entries are the eigenvalues of $A$, then
   $$
    R_z(D_A)=\sum_{j=1}^n\frac{Q_j}{\la_j(A)-z},
    $$
    where $Q_k=(.,e_k)e_k$. 
Similarly, 
   $$
    R_z(D_B)=\sum_{j=1}^n\frac{Q_j}{\la_j(B)-z}.
    $$
Taking into account that
 $Q_{s}V_A Q_{m}=0,\;Q_{s}V_B Q_{m}=0\;(s\geq m)$, we get
$$
  C_{jk}=
\sum_{1\leq s_1<s_2< \ldots < s_{j+1}<m_1<m_2< \ldots < m_{k+1}<n}
Q_{s_1}V_AQ_{s_2} V_A \cdots
$$
$$
V_AQ_{s_{j+1}}Q_{m_1}V_BQ_{m_2} V_B \ldots V_B Q_{m_{k+1}}
J(s_1,   \ldots , s_{j+1}, m_1,   \ldots , m_{k+1}),
$$
where
$$
J(s_1,   \ldots, s_{j+1}, m_1,   \ldots m_{k+1})=
$$
$$
\frac{(-1)^{k+j}}{4\pi^2}    \int_{L_A}\int_{L_B}\frac{f(z, w)dz\;dw}
{(\la_{s_1}(A)-z) \cdots( \la_{s_{k+1}}(A)-z)(\la_{m_1}(B)-w) \cdots  ( \la_{m_{k+1}}(B)-w)}.
    $$
    Lemma 2.5.1 from \cite{gi03} gives us the estimate
    $$
    \|C_{jk}\| \leq \max_{1\leq s_1<\ldots < s_{j+1}
< m_1<\ldots < m_{k+1}< n} 
|J(s_1,   \ldots, s_{j+1}, m_1,   \ldots m_{k+1})|\|\;|V_A |^j\;|V_B |^k\;\|.
    $$
    Due to Lemma 2.1,
$$
|J(s_1, \ldots, s_{j+1}, m_1, \ldots, m_{k+1})|\leq
\sup_{z\in co(A), w\in co(B)}
\frac{|f^{(j,k)}(z, w)|}{j!k!}.
$$
This inequality and (2.5) imply the required result. 
\end{proof}

{\em Proof of Theorem 1.1}: Theorem 2.5.1 from \cite{gi03} implies
    
$$
    \|V^{k}\;\|\leq \et_k k! N^k(V) \;
\eq 2.6)   
 $$
for any nilpotent matrix $V\in M(\cn)$.
Take into account that $N(|V_A|)=N(V_A)$. 
Moreover, thanks to Lemma 2.3.2 from \cite{gi03}, $N(V_A)=g(A)$. Thus
    $$
    \|\;|V_A|^{k}\;\|\leq k!\et_{k} g^k(A) \;(k= 1,..., n-1).
\eq 2.7)  
  $$
The similar inequality holds for $V_B$.
Now the previous lemma  yields the required result. $\Box$

\section{Functions of  matrices on tensor products}  

Let $E_1=\bc ^{n_1}, E_2=\bc ^{n_2},$ be the Euclidean spaces of the 
dimensions $n_1$ and $n_2$, 
 with  the scalar products
$<.,.>_1$ and $<.,.>_2$, respectively, and  
the norms $\|.\|_l=\sqrt{<.,.>_l}\;(l=1,2)$.
 Let $H=E_1\otimes E_2$ be the tensor product of 
$E_1$ and $E_2$ with the scalar product
defined by
$$
<y\otimes h,y_{1}\otimes h_{1}>_H
\equiv <y,y_{1}>_{1}\;<h,h_1>_{2 } \;\;
(y, y_{1}\in E_1;\;h, h_{1} \in E_2)
$$
and  the cross norm $\|.\|_H=\sqrt{<.,.>_H}$, cf. \cite{ma}.
In addition, $I=I_H$ and $I_l$  mean the unit operators in $H$ and $E_l$, respectively. 
So $H=\cn\mbox{ with }n=n_1 n_2$.

Recall that  $M(E)$  is the set of all linear operators in a space $E$.
In  this section it is assumed that $A\in M(E_1)$ and $B\in M(E_2)$.

Again let $\Om_A$ and $\Om_B$ 
be open simple connected sets containing  $\si(A)$ and $\si(B)$, respectively.
Let $f$ be a scalar function analytic on $\Om_A\times\Om_B$. 
Introduce the operator valued function
$$
f(A, B):=-\frac{1}{4\pi^2}\int_{L_B} \int_{L_A} f(z, w) R_{z}(A)\otimes  R_{w}(B) 
dw\;dz,
\eq 3.1)
$$
where  $L_A\subset \Om_A, L_B\subset \Om_B$ 
are closed contour surrounding $\si(A)$ and $\si(B)$, respectively.
If the series 
$$
f(z, w)=\bs 0 \8  \bsj 0 \8  c_{jk} z^{j}w^k
$$
converges for $|z|\leq r_s(A), |w|\leq r_s(B)$, where
$r_s(A)$ is  the spectral radius of $A$, 
then (3.1) holds. Besides,
$$
f(z, w)=\bs 0 \8  \bsj 0 \8  c_{jk} A^{j}\otimes B^k.
$$

\begin{theorem} Let $A\in M(E_1)$ and $B\in M(E_2)$ 
and $f(z,w)$ be regular on a neighborhood of $co(A)\times co(B)$. Then
 $$
 \|f(A, B)\|_H\leq \sum_{ j=0}^{n_1-1}\bs 0 {n_2-1} 
\et_j\et_k g^j(A) g^k(B)
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|.
    $$
 \end{theorem}

\begin{proof} 
Put $\ti A=A\otimes I_2$, $\ti B=I_1\otimes B_2$.
Now Lemma 2.2 implies
$$
\|f(A, B)\|_H\leq \sum_{j,k= 0}^{j+k \leq  n-1}
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|
\frac{\|\;|V_{\ti A}|^j \;|V_{\ti B}|^k\|_H}{j!k!}
$$
where $V_{\ti A}, V_{\ti B}$ are the nilpotent parts of $\ti A$ and $\ti B$,
respectively. But
$$
V^{n_1}_{\ti A}=V^{n_1}_{A}\otimes I_2=0.
$$
Similarly, $V^{n_2}_{\ti B}=0$. Thus,
$$
\|f(A, B)\|_H\leq \bs 0 {n_2} \sum_{j= 0}^{n_1}
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|
\frac{\|\;|V_{A}|^j\|_1 \;\|\;|V_{B}|^k\|_2}{j!k!}
$$
Now the required result follows from  (2.7). 
\end{proof}

Taking into account (1.3), we get

\begin{cor} Under the hypothesis of {\rm Theorem 3.1},
the estimate
$$
\|f(A, B)\|_H\leq \bs 0 {n_2-1}\sum_{j= 0}^{n_1-1}
\frac{g^j(A) g^k(B)}{(j!k!)^{3/2}}
\sup_{z\in co(A), w\in co(B)}|f^{(j,k)}(z, w)|
$$
is true.
\end{cor}

\newpage
{\bf Acknowledgment.} 
I am very grateful to the referee for very helpful remarks.

\begin{thebibliography}{11}
\bibitem{ar}
R. Arens and A.P. Calderon. Analytic functions of several Banach
algebra elements. {\em Ann. Math.}, 62:204--216, 1955.  
\bibitem{da}
Yu L. Daleckii and M.G. Krein. {\em Stability of Solutions
of Differential Equations in Banach Space}. Amer. Math. Soc., Providence, R.I., 1974.
\bibitem{ges}
M.I. Gel'fand and G.E. Shilov. {\em Some Questions of Theory of
Differential Equations}. Nauka, Moscow (in  Russian), 1958.
\bibitem{ge}
A.O. Gel'fond. {\em Calculations of Finite Differences}. Nauka, Moscow 
(in Russian), 1967.
\bibitem{gi93}
M.I. Gil'. Estimates for norm of matrix-valued functions,
{\em Linear and Multilinear Algebra}, 35:65--73, 1993.
\bibitem{gi98}
M.I. Gil'. {\em Stability of Finite and Infinite Dimensional Systems}.
Kluwer Academic Publishers, Boston-Dordrecht-London, 1998.
\bibitem{gi03}
M.I. Gil'. {\em Operator Functions and Localization of Spectra}. Lectures Notes 
In Mathematics vol. 1830, Springer-Verlag, Berlin, 2003. 
\bibitem{ho}
R.A. Horn and C.R. Johnson. {\em Topics in Matrix Analysis}. Cambridge
University Press, Cambridge, 1991.
\bibitem{ma}
M. Marcus.  
{\em Finite Dimensional Multilinear Algebra}, Vol. I, Marcel Dekker, 
New York, 1973. 
%\bibitem{ma1}
%M. Marcus and H. Minc. {\em A Survey of Matrix Theory
%and Matrix Inequalities}. Allyn and Bacon, Boston, 1964 .
\bibitem{mu}
V. M\"uller. {\em Spectral Theory of Linear Operators}.
Birkh\"ausr Verlag, Basel, 2003.   
\bibitem{os}
A.M. Ostrowski. {\em Solution of Equations in Euclidean and Banach
spaces}.  Academic Press, New York - London, 1973.
\bibitem{ta}
J.L. Taylor. Analytic functional calculus for several commuting operators.
{\em Acta Math.}, 125:1--38, 1970.
\end{thebibliography}
\end{document}

