Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Free Noncommutative Functions:
An Introduction
Dmitry Kaliuzhnyi-Verbovetskyi1
(Drexel University)
Iowa–Nebraska Functional Analysis Seminar, Des Moines IA,
April 19, 2014
1
A joint work with V. Vinnikov
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Noncommutative (nc) multi-operator spectral theory: J. L. Taylor
Free probability: D.-V. Voiculescu...
NC free semi-algebraic geometry: J. W. Helton, M. Putinar, S.
McCullough, I. Klep...
Representations of nc disk algebras, free holomorphic functions on
nc domains: G. Popescu
Representations of tensor algebras over C ∗ correspondences: P.
Muhly and B. Solel,
Noncommutative functions: J. Agler, J. E. McCarthy, N. Young ...
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Let R be a commutative ring, and let M be a module over R. We
define the nc space over M,
Mnc =
∞
a
Mn×n .
n=1
For X ∈ Mn×n and Y ∈ Mm×m their direct sum is
X 0
X ⊕Y =
∈ M(n+m)×(n+m) .
0 Y
A subset Ω ⊆ Mnc is called a nc set if it is closed under direct
sums: denoting Ωn = Ω ∩ Mn×n , we have
X ∈ Ωn , Y ∈ Ωm =⇒ X ⊕ Y ∈ Ωn+m .
A nc set Ω is called right admissible if for every X ∈ Ωn , Y ∈ Ωm ,
and Z ∈ Mn×m there exists an invertible r ∈ R such that
X rZ
∈ Ωn+m .
0 Y
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Examples
Right admissible nc sets:
1. The set of upper triangular matrices over M.
2. The set of matrices X ∈ Cnc with σ(X ) ⊆ S, for a given
S ⊆ C. In particular, the set of nilpotent matrices (for
S = {0}); the set of invertible matrices (for S = C \ {0}).
3. The nc unit ball Bnc (0, 1) = {X ∈ Rnc : kX k2,2 < 1}.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Notice that matrices over R act from the right and from the left
on matrices over M by the standard rules of matrix multiplication:
if X ∈ Mp×q and T ∈ Rr ×p , S ∈ Rq×s , then
TX ∈ Mr ×q ,
XS ∈ Mp×s .
In the case of M = Rd , using the identification
p×q
d
∼
Rd
= Rp×q ,
d
we have, for d-tuples X = (X1 , . . . , Xd ) ∈ (Rn×n ) and
d
Y = (Y1 , . . . , Yd ) ∈ (Rm×m ) ,
d
X
0
X1 0
X ⊕Y =
,..., d
∈ R(n+m)×(n+m) ;
0 Y1
0 Yd
d
and for a d-tuple X = (X1 , . . . , Xd ) ∈ (Rp×q ) and matrices
T ∈ Rr ×p , S ∈ Rq×s ,
d
d
TX = (TX1 , . . . , TXd ) ∈ Rr ×q , XS = (X1 S, . . . , Xd S) ∈ Rp×s .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
When R = K, K = R or C, and thus M = V is a vector space,
consider the following three topologies on the nc space Vnc :
1. Finitely-open topology. A set Ω ⊆ Vnc is finitely-open if for
every n ∈ N and every finite-dimensional subspace X ⊆ V n×n the
set Ωn ∩ X is open in the relative topology of X .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
2. Norm topology. Suppose each V n×n is a Banach space in a
norm k · kn , and the system of norms k · kn , n ∈ N, is admissible:
I
For every n, m ∈ N there exist C1 (n, m), C10 (n, m) > 0 such
that for all X ∈ V n×n and Y ∈ V m×m ,
C1 (n, m)−1 max{kX kn , kY km } ≤ kX ⊕ Y kn+m
≤ C10 (n, m) max{kX kn , kY km }.
I
For every n ∈ N there exists C2 (n) > 0 such that for all
X ∈ V n×n and S, T ∈ Kn×n ,
kSXT kn ≤ C2 (n)kSk kX kn kT k,
where k · k denotes the operator norm of Kn×n with respect to
the standard Euclidean norm of Kn .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Equivalently, injections,
ιij : V → V n×n ,
ιij : v 7→ Eij v ,
πij : V n×n → V,
πij : X 7→ Xij ,
and projections,
are bounded for all n, i, j.
(When C1 , C10 , C2 are independent of n, m, all ιij and πij are
uniformly completely bounded.)
A set Ω ⊆ Vnc is open if for every n ∈ N the set Ωn ⊆ V n×n is
open in norm k · kn .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
3. Uniform topology. A vector space V is called an operator
space if Vnc is equipped with an admissible system of norms k · kn
on V n×n , n ∈ N, and C1 (n, m) = C10 (n, m) = C2 (n) = 1 for all
n, m ∈ N. In particular, all injections ιij are complete isometries
and all projections πij are complete co-isometries.
The nc ball centered at Y ∈ V s×s of radius is
∞ n
m
a
M
sm×sm Bnc (Y , ) :=
X ∈V
: X −
Y
m=1
α=1
sm
NC balls form a base for the uniform topology on Vnc .
o
< .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Let M and N be modules over a unital commutative ring R, and
let Ω ⊆ Mnc be a nc set. A mapping f : Ω → Nnc with
f (Ωn ) ⊆ N n×n is called a nc function if
I
f respects direct sums:
f (X ⊕ Y ) = f (X ) ⊕ f (Y ),
I
X , Y ∈ Ω.
(1)
f respects similarities: if X ∈ Ωn and S ∈ Rn×n is invertible
with SXS −1 ∈ Ωn , then
f (SXS −1 ) = Sf (X )S −1 .
(2)
Proposition
A mapping f : Ω → Nnc with f (Ωn ) ⊆ N n×n respects direct sums
and similarities, i.e., (1) and (2) hold iff f respects intertwinings:
for any X ∈ Ωn , Y ∈ Ωm , and T ∈ Rn×m such that XT = TY ,
f (X )T = Tf (Y ).
(3)
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Examples
NC functions.
1. NC polynomials viewed as functions on matrices, e.g.,
p(X ) = X1 X2 − X2 X1 ,
X = (X1 , X2 ) ∈ (R2 )nc ,
so that p : (R2 )nc → Rnc . In general,
X
p(X ) =
pw X w , X = (X1 , . . . , Xd ) ∈ (Rd )nc ,
w ∈Gd : |w |≤N
where Gd is the free monoid on d generators g1 , . . . , gd , with
the unit element ∅ (the empty word); X w = Xi1 · · · Xik for a
word w = gi1 · · · gik (and X ∅ = I ); the length of the word
w = gi1 · · · gik is |w | = k (and |∅| = 0); pw ∈ R for every
w ∈ Gd . So, p : (Rd )nc → Rnc .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
2. NC power series evaluated on matrices:
X
f (X ) =
fw X w , X = (X1 , . . . , Xd ) ∈ Ω,
w ∈Gd
where Ω ⊆ (Kd )nc is a nc set where the series converges (in
some sense), K = R or K = C, thus f : Ω → Knc .
3. NC rational expressions, e.g., realization formulas
f (X ) = D(X ) + C (X )(I − A(X ))−1 B(X ),
where A(X ), B(X ), C (X ), D(X ) are linear functions, or
perhaps even higher degree nc polynomials. See [J. A. Ball, G.
Groenewald, and T. Malakorn].
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
Let f : Ω → Nnc be a nc function on a right admissible nc set Ω.
X ∈ Ωn , Y ∈ Ωm , and Z ∈ Mn×m be such that
Let
X Z ∈Ω
n+m . Then
0 Y
X
f
0
Z
Y
f (X ) ∆R f (X , Y )(Z )
=
,
0
f (Y )
where the off-diagonal block entry ∆R f (X , Y )(Z ) as a function of
Z has a unique extension to a linear function on Mn×m .
Theorem
f (X ) − f (Y ) = ∆R f (Y , X )(X − Y )
= ∆R f (X , Y )(X − Y ), n ∈ N, X , Y ∈ Ωn .
∆R is called the right nc difference-differential operator.The linear
mapping ∆R f (Y , Y )(·) plays the role of a nc differential.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Under appropriate continuity conditions, ∆R f (Y , Y )(Z ) is the
directional derivative of f at Y in the direction Z .
In the case where M = Rd , the finite difference formula turns into
f (X ) − f (Y ) =
N
X
∆R,i f (Y , X )(Xi − Yi ),
X , Y ∈ Ωn ,
i=1
with the right partial difference-differential operators ∆R,i :
∆R,i f (Y , X )(C ) := ∆R f (Y , X )(0, . . . , 0, |{z}
C , 0, . . . , 0).
i th place
The left nc full and partial difference-differential operators ∆L ,
∆L,i , i = 1, . . . , d, are defined analogously.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
For any X ∈ Ωn , Y ∈ Ωm , T ∈ Rn×m , n, m ∈ N:
f (X )T − Tf (Y ) = ∆R f (Y , X )(XT − TY ).
Proof. Let r ∈ R be invertible and such that
X r (TY − XT )
∈ Ωn+m .
0
Y
Then
Then
i.e.,
X
0
X
f
0
r (TY − XT )
Y
rT
rT
=
Y.
Im
Im
r (TY − XT )
rT
rT
=
f (Y ),
Y
Im
Im
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
f (X ) ∆R f (X , Y )(r (TY − XT )) rT
rT
=
f (Y ).
0
f (Y )
Im
Im
Comparing the 1st block entries in the matrix products on the
right-hand side and on the left-hand side, we obtain
rf (X )T + r ∆R f (X , Y )(TY − XT )) = rTf (Y ).
Cancelling r and rearranging, we obtain
f (X )T − Tf (Y ) = ∆R f (X , Y )(XT − TY ).
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
As a function of X and Y , ∆R f (X , Y )(·) respects direct sums and
similarities, or equivalently, respects intertwinings: if X ∈ Ωn ,
e ∈ Ωñ , Y
e ∈ Ωm̃ , and T ∈ Rn×ñ , S ∈ Rm×m̃ are such
Y ∈ Ωm , X
that
e , YS = S Y
e,
XT = T X
and Z ∈ Men×m , then
e, Y
e )(ZS).
∆R f (X , Y )(TZ )S = T ∆R (X
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Example
Let f : Rnc → Rnc , f (X ) = X 2 . Since
X
f
0
Z
Y
X
=
0
Z
Y
2
2
X XZ + ZY
=
,
0
Y2
e , YS = S Y
e , then
∆R f (X , Y )(Z ) = XZ + ZY . If XT = T X
∆R f (X , Y )(TZ )S = (XTZ + TZY )S = XTZS + TZYS
e ZS + TZS Y
e = T (X
e ZS + ZS Y
e ) = T ∆R f (X
e, Y
e )(ZS).
= TX
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
We denote the class of functions h on Ω × Ω whose values on
Ωn × Ωm are linear mappings Mn×m → N n×m satisfying the
property above (with ∆R f replaced by h) as
T 1 = T 1 (Ω; Nnc , Mnc ).
Thus for a nc function f , ∆R f ∈ T 1 .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
More generally, we define the class of nc functions of order k,
T k = T k (Ω; N0,nc , N1,nc , . . . , Nk,nc )
as a class of functions on Ωk+1 , where Ω ⊆ Mnc is a nc set, whose
values on Ωn0 × · · · × Ωnk are k-linear forms
N1 n0 ×n1 × · · · × Nk nk−1 ×nk → N0 n0 ×nk ,
and which respect direct sums and similarities, or equivalently,
e j ∈ Ωen , Tj ∈ Rnj ×enj satisfy
respect intertwinings: if X j ∈ Ωnj , X
j
ej,
X j Tj = Tj X
j = 0, 1, . . . , k,
and Z j ∈ Menj ×nj+1 , j = 1, . . . , k, then
f (X 0 , . . . , X k )(T0 Z 1 , . . . , Tk−1 Z k )Tk
e 0, . . . , X
e k )(Z 1 T1 , . . . , Z k Tk ).
= T0 f ( X
T 0 = T 0 (Ω; Nnc ) is the class of nc functions f : Ω → Nnc .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Example
Let f0 , . . . , fk ∈ T 0 (Rnc ; Rnc ). We define
f ∈ T k (Rnc ; Rnc , . . . , Rnc ) by
f (X 0 , . . . , X k )(Z 1 , . . . , Z k ) = f0 (X 0 )Z 1 f1 (X 1 )Z 2 · · · Z k fk (X k ).
Suppose that
ej,
X j Tj = Tj X
j = 0, 1, . . . , k.
Then
f (X 0 , . . . , X k )(T0 Z 1 , . . . , Tk−1 Z k )Tk
= f (X 0 )T0 Z 1 f1 (X 1 )T1 Z 2 · · · Tk−1 Z k fk (X k )Tk
e 0 )Z 1 T1 f (X
e 1 )Z 2 · · · Z k Tk f (X
ek)
= T0 f0 (X
e k )(Z 1 T1 , . . . , Z k Tk ).
e 0, . . . , X
= T0 f (X
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
We define ∆R : T k → T k+1 as follows:
f
k0
h
i
X
Z
1
k−1
k0
k00
X ,...,X
,
Z
,
.
.
.
,
Z
,
row
Z
,
Z
0 X k00
h = row f X 0 , . . . , X k−1 , X k0 Z 1 , . . . , Z k−1 , Z k0 ,
∆R f X 0 , . . . , X k−1 , X k0 , X k00 Z 1 , . . . , Z k−1 , Z k0 , Z
i
+f X 0 , . . . , X k−1 , X k00 Z 1 , . . . , Z k−1 , Z k00 .
0
k−1
Notice, that ∆R = k ∆R . We define j ∆R similarly for
j = 0, . . . , k − 1.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
Let f ∈ T 0 (Ω; Nnc ). Then
∆`R f (X 0 , . . . , X ` )(Z 1 , . . . , Z ` )

X0 Z1

 0 X 1


..
= f  ...
.

 ..
 .
0 ···
0
..
.
..
.
..
.
···
···
..
.
..
.
X `−1
0

0

.. 

. 


.

0


Z ` 
X`
1,`+1
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Generalized finite difference formulas:
f X 0 , . . . , X k−1 , X Z 1 , . . . , Z k T
− f X 0 , . . . , X k−1 , Y
Z 1, . . . , Z k T
= ∆R f X 0 , . . . , X k−1 , X , Y
Z 1 , . . . , Z k , XT − TY ,
f (X , X 1 , . . . , X k )(TZ 1 , . . . , Z k )−Tf (Y , X 1 , . . . , X k )(Z 1 , Z 2 , . . . , Z k )
= 0 ∆R f (X , Y , X 1 , . . . , X k )(XT − TY , Z 1 , . . . , Z k ),
f (X 0 , . . . , X j−1 , X , X j+1 , . . . , X k )(Z 1 , . . . , Z j , TZ j+1 , . . . , Z k )
−f (X 0 , . . . , X j−1 , Y , X j+1 , . . . , X k )(Z 1 , . . . , Z j−1 , Z j T , Z j+1 , . . . , Z k )
= j ∆R f (X 0 , . . . , X j−1 , X , Y , X j+1 , . . . , X k )
(Z 1 , . . . , Z j , XT − TY , Z j+1 , . . . , Z k ),
0 < j < k.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
We use the calculus of higher order nc difference-differential
operators to derive a nc analogue of Taylor’s formula, which we
call the Taylor–Taylor (TT) formula.
Theorem
Let f ∈ T 0 (Ω; Nnc ) with Ω ⊆ Mnc a right admissible nc set,
s ∈ N, and Y ∈ Ωs . Then for every m ∈ N, N ∈ Z+ , and X ∈ Ωsm ,
f (X ) =
N
X
∆`R f
`=0
+ ∆N+1
f
R
m
M
|α=1
m
M
Y,...,
|α=1
Y,...,
m
M
Y
X−
α=1
{z
`+1 times
m
M
Y,X
N+1 times
m
M
Y,...,X −
α=1
}
X−
|
m
M
}
|
m
M
}
` times
Y,...,X −
m
M
Y .
α=1
{z
Y
α=1
{z
α=1
α=1
{z
N+1 times
}
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
In the case where V = Rd , we obtain
f (X ) =
N
X
X
>
∆w
R f
m
M
`=0 w =gi1 ···gi`
|α=1
Y,...,
m
M
Y
α=1
{z
`+1 times
m
M
Xi1 −
}
Yi1 , . . . , Xi` −
α=1
+
X
>
∆w
R f
m
M
w =gi1 ···giN+1
Y,...,
|α=1
Xi1 −
m
M
m
M
Yi`
α=1
m
M
Y,X
α=1
{z
N+1 times
}
Yi1 , . . . , XiN+1 −
α=1
>
m
M
α=1
where for a word w = gi1 · · · gi` , ∆w
R := ∆R,i` · · · ∆R,i1 .
YiN+1 ,
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
If Y = (µ1 Is , . . . , µd Is ), this reduces to a genuine nc power
expansion
f (X ) =
N X
X
`=0 |w |=`
>
(X − µIsm )w ∆w
. . . , µ)
R f (µ,
| {z }
`+1 times
+
X
>
(X − µIsm )w ∆w
. . . , µ , X ).
R f ( µ,
| {z }
|w |=N+1
N+1 times
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
1. One can interpret the `-linear mappings
∆`R f (Y , . . . , Y ) : Ms×s
`
→ N s×s
as linear mappings
⊗`
∆`R f (Y , . . . , Y ) : Ms×s
→ N s×s ,
and then extend them to matrices over the tensor algebra
T(Ms×s ) as acting entrywise. So,
∆`R f
m
M
α=1
Y,...,
m
M
Y
X−
α=1
m
M
Y,...,X −
α=1
= X−
m
M
Y
α=1
m
M
α=1
Y
s `
∆`R f (Y , . . . , Y ).
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Here the “faux” product s is understood as a product in
m×m
T(Ms×s ), i.e., for Z 0 , Z 00 ∈ Msm×sm ∼
,
= (Ms×s )
(Z 0 s Z 00 )ij =
m
X
Zik0 ⊗ Zkj00 .
k=1
For Y ∈ Ms×s , let Nilp(Ms×s , Y ) denote the nc set of matrices
s `
m
∞
L
`
= 0 for some
Y
Msm×sm such that X −
X ∈
α=1
m=1
` ∈ N.Then every nc function f : Nilp(Ms×s , Y ) → N s×s
has
nc
the Taylor–Taylor series expansion
f (X ) =
∞ X
`=0
X−
m
M
α=1
Y
s `
∆`R f (Y , . . . , Y ).
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
Given a nc function f : Mnc → Nnc , for every n, m ∈ N,
T ∈ Rsn×sm , and j = 0, . . . , ` − 1, the `-linear mappings
s×s )` → N s×s , ` = 0, 1, . . . , satisfy
f` =∆`R f (Y , . . ., Y ) : (M
f` Z 1 , . . . , Z k T − f` Z 1 , . . . , Z k T
n
m
M
M
= f`+1 Z 1 , . . . , Z k ,
Y T −T
Y ,
α=1
1
k
1
β=1
k
f` (TZ , . . . , Z ) − Tf` (Z , . . . , Z )
n
m
M
M
= f`+1
Y T −T
Y , Z 1, . . . , Z k ,
α=1
1
j
j+1
β=1
k
f` (Z , . . . , Z , TZ , . . . , Z )−f` (Z , . . . , Z j−1 , Z j T , Z j+1 , . . . , Z k )
n
m
M
M
1
j
= f`+1 Z , . . . , Z ,
Y T −T
Y , Z j+1 , . . . , Z k , 0 < j < k.
α=1
1
β=1
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Conversely, given a sequence of `-linear mappings
`
f` : (Ms×s ) → N s×s , ` = 0, 1, . . . , satisfying
theconditions
above, the mapping f : Nilp(Ms×s , Y ) → N s×s
defined by
nc
f (X ) =
∞ X
`=0
X−
m
M
Y
s `
f`
α=1
is a nc function and f` = ∆`R f (Y , . . . , Y ) are the Taylor–Taylor
coefficients of f .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Not every nc power series centered at Y defines a nc function on
Nilp(Ms×s , Y )!
Example
Let Y ∈ C2×2 . In general,
m
m
m
m
2
M
M
M
M
Y2
Y = X2 − X
Y −
Y X+
p(X ) = X −
α=1
α=1
α=1
is not a nc function: it does not respect similarities.
α=1
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
2.
Theorem
Let R = K be an infinite field, and let f : (Kd )nc → Knc be a nc
function such that for every n ∈ N each matrix entry of
f (X1 , . . . , Xd ) is a polynomial in (Xi )j,k , i = 1, . . . , d,
j, k = 1, . . . , n, of a uniformly (in n) bounded degree. Then f is a
nc polynomial, i.e.,
X
f (X ) =
fw X w .
w ∈Gd ,|w |≤N
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Proof Let f (X1 , . . . , Xd ) be a polynomial in (Xi )j,k , i = 1, . . . , d,
j, k = 1, . . . , n, of total degree N. For Y = (0, . . . , 0) ∈ Kd , write
N
X
X
>
f (X ) =
X w ∆w
R f (0, . . . , 0)
`=0 w ∈Gd : |w |≤N
+
X
>
X w ∆w
R f (0, . . . , 0, X ).
|w |=N+1
Since for w = gi1 · · · giN+1 ,

0

>

X w ∆w
R f (0, . . . , 0, X ) = f 
ei1 ⊗ Xi1
..
.

..
.
0
eiN+1 ⊗ XiN+1
X



1,N+2
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
is a polynomial in (Xi )jk of total degree at least N + 1, we must
have
X
>
X w ∆w
R f (0, . . . , 0, X ) = 0,
|w |=N+1
so that f is a nc polynomial.
The assumption of boundedness of degrees cannot be omitted!
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Example
Let {pn }n∈N be a sequence of homogeneous nc polynomials
2
vanishing on (Kn×n ) . E.g.,
X
π(1)−1
π(n+1)−1
pn =
sign(π)x1
x2 · · · x1
x2 .
π∈Sn+1
αn := deg pn =
(n+1)(n+2)
.
2
Then
f (X1 , X2 ) =
∞
X
pn (X1 , X2 )
n=0
is a nc function which is a polynomial in matrix entries for each n.
However, f is not a nc polynomial.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
Let f be a nc function on (Kd )nc , where K is an infinite field, with
values in Nnc . Assume that for each n, f (X1 , . . . , Xd ) is a
polynomial function of degree Ln in dn2 commuting variables
(Xi )jk , i = 1, . . . , d; j, k = 1, . . . , n, with values in N n×n . Then
there exists a unique sequence of homogeneous nc polynomials
fj ∈ N hx1 , . . . , xd i of degree j, j = 0, 1, . . . , such that for all
n ∈ N,
d
I
fj vanishes on (Kn×n ) for all j > Ln ,
I
fLn does not vanish identically on (Kn×n ) ,
d
d
and for X ∈ (Kn×n ) ,
f (X ) =
∞
X
j=0
fj (X ) =
Ln
X
j=0
fj (X ).
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Let V be a vector space over C (resp., a Banach space equipped
with an admissible system of matrix norms over V, operator
space), let W be a Banach space equipped with an admissible
system of matrix norms over W, and let Ω ⊆ Vnc be a finitely-open
(resp., norm-open, uniformly open) nc set.
Facts. 1. If a nc function f : Ω → Wnc is locally bounded on
slices, i.e., for every Y ∈ Ωn and Z ∈ V n×n , f (Y + λZ ) is bounded
for |λ| < , then f is analytic on slices, i.e., for every Y ∈ Ωn and
Z ∈ V n×n , f (Y + λZ ) is analytic as a function of λ.
2. If a nc function f : Ω → Wnc is locally norm-bounded, then f is
analytic (= locally norm-bounded + Gâteaux differentiable).
3. If a nc function f : Ω → Wnc is locally uniformly bounded, i.e.,
for every Y ∈ Ω, f is bounded in some nc ball Bnc (Y , ), then f is
uniformly analytic (= locally uniformly bounded + Gâteaux
differentiable).
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
E.g., in the uniform topology setting we have
Theorem
Let a nc function f : Ω → Wnc be uniformly locally bounded. For
s ∈ N, Y ∈ Ωs , let δ := sup{r > 0 : f is bdd on Bnc (Y , r )}. Then
f (X ) =
∞ X
`=0
X−
m
M
α=1
Y
s `
∆`R f (Y , . . . , Y )
| {z }
`+1 times
holds, with the TT series convergent absolutely and uniformly on
every open nc ball Bnc (Y , r ) with r < δ.
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Proof.
. For some
Let r < δ and kf (Z )ksmZ ≤ M for all Z ∈ Bnc Y , r +δ
2
X ∈ Bnc Y , r , set LmX

LmX
r +δ
α=1


Z := 


Y
2r
X−
..
α=1
.
Y
..
.
..
.
r +δ
2r
LmX
X − α=1
LmX
α=1 Y



 .
Y 
L(`+1)m
Since kZ − β=1 X Y ksmX (`+1) ≤ r +δ
2 , we have
m
2r `
M
s `
Y
∆`R f (Y , . . . , Y )
= kf (Z )1,`+1 ksmX
X−
| {z } smX
r +δ
α=1
`+1 times
≤M
2r `
.
r +δ
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
Let Ω ⊆ (Cd )nc be uniformly-open, W an operator space, and a nc
function f : Ω → Wnc uniformly locally bounded. For every s ∈ N,
Y ∈ Ωs , let δ := sup{r > 0 : f is bdd on Bnc (Y , r )}. Then
m
s w
M
X >
,...,Y)
∆w
X−
Y
f (X ) =
R f (Y
| {z }
w ∈Gd
α=1
`+1 times
with the TT series convergent absolutely and uniformly on every
open nc diamond about Y of radius r < δ,
∞ n
d m
o
a
M
X
♦nc (Y , r ) :=
X ∈ Ωsm :
Yj < r .
Xj −
m=1
j=1
α=1
sm
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Let V and W be operator spaces. Let f` : V ` → W be a sequence
of completely bounded `-linear functions (in the sense of
Christensen and Sinclair), that is,
kf` kcb := sup kf` (Z 1 , . . . , Z ` )k < ∞,
where the sup is taken over all n0 , . . . , n` ∈ N and all matrices
Z 1 ∈ V n0 ×n1 , . . . , Z ` ∈ V n`−1 ×n` of norm 1. Equivalently, the
linear mappings f` : V ⊗` → W are completely bounded.
Let Y ∈ V s×s for some s ∈ N. For the series
X
X−
m
M
Y
s `
f`
α=1
`
we define the Cauchy–Hadamard radius
ρcb =
lim
`→∞
−1
p
`
kf` kcb
.
(4)
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
Theorem
The series (4) converges uniformly and absolutely on every nc ball
Bnc (Y , r ) with r < ρcb . Moreover, the convergence is normal, i.e.,
∞
X
sup
`=0 W ∈Bnc (Y ,r )
mW
s ` M
f` Y
W−
α=1
smW
< ∞.
The series (4) fails to converge uniformly on every nc ball
Bnc (Y , r ) with r > ρcb .
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
For more detail, see arXiv:1212.6345
Historic remarks NC sets and nc functions Higher order nc functions Applications of the TT formula NC power series
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