Noncommutative functions: Algebraic and analytic results Dmitry - - PowerPoint PPT Presentation

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results

Noncommutative functions: Algebraic and analytic results

Dmitry Kaliuzhnyi-Verbovetskyi1

Department of Mathematics Drexel University (Philadelphia, PA)

October 3, 2010 Bill Helton Workshop, UCSD

1A joint work with V. Vinnikov Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

For a vector space V over a field K, we define the noncommutative (nc) space over V Vnc =

∞

• n=1

Vn×n. For X ∈ Vn×n and Y ∈ Vm×m we define their direct sum X ⊕ Y = X Y

• ∈ V(n+m)×(n+m).

Notice that matrices over K act from the right and from the left

• n matrices over V by the standard rules of matrix multiplication:

if X ∈ Vp×q and T ∈ Kr×p, S ∈ Kq×s, then TX ∈ Vr×q, XS ∈ Vp×s. A subset Ω ⊆ Vnc is called a nc set if it is closed under direct sums; explicitly, denoting Ωn = Ω ∩ Vn×n, we have X ⊕ Y ∈ Ωn+m for all X ∈ Ωn, Y ∈ Ωm.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

In the case of V = Kd we identify matrices over V with d-tuples of matrices over K:

• Kdp×q ∼

=

• Kp×qd .

Under this identification, for d-tuples X = (X1, . . . , Xd) ∈ (Kn×n)d and Y = (Y1, . . . , Yd) ∈ (Km×m)d, X ⊕ Y = X1 Y1

• , . . . ,

Xd Yd

• ∈
• K(n+m)×(n+m)d

; and for a d-tuple X = (X1, . . . , Xd) ∈ (Kp×q)d and matrices T ∈ Kr×p, S ∈ Kq×s, TX = (TX1, . . . , TXd) ∈

• Kr×qd ,

XS = (X1S, . . . , XdS) ∈

• Kp×sd .

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

Let V and W be vector spaces over K, and let Ω ⊆ Vnc be a nc

• set. A mapping f : Ω → Wnc with f (Ωn) ⊆ Wn×n is called a nc

function if f satisfies the following two conditions:

◮ f respects direct sums:

f (X ⊕ Y ) = f (X) ⊕ f (Y ), X, Y ∈ Ω. (1)

◮ f respects similarities: if X ∈ Ωn and S ∈ Kn×n is invertible

with SXS−1 ∈ Ωn, then f (SXS−1) = Sf (X)S−1. (2)

Proposition

A mapping f : Ω → Wnc with f (Ωn) ⊆ Wn×n respects direct sums and similarities, i.e., (1) and (2) hold iff f respects intertwinings: for any X ∈ Ωn, Y ∈ Ωm, and T ∈ Kn×m such that XT = TY , f (X)T = Tf (Y ). (3)

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

(a) NC polynomials and nc rational expressions. In many engineering applications, matrices appear as natural

• variables. Stability problems in control theory are usually reduced

to Stein, Lyapunov, or Riccati equations or inequalities where the left-hand side is a nc polynomial, e.g., as in the continuous-time Riccati inequality p(X, A, A∗, B, C) := AX + XA∗ + XBX + C ≤ 0,

• r a nc rational expression, as in the discrete-time Riccati inequality

r(X, A, A∗, B, B∗, C, C ∗, D, D∗) := A∗XA − X + C ∗C + (C ∗D + A∗XB)(I − D∗D − B∗XB)−1(D∗C + B∗XA) ≤ 0. Other polynomial and rational matrix inequalities arise in

• ptimization and related problems, such as nc sum-of-squares

(SoS) representations of positive nc polynomials, factorization of hereditary polynomials, nc positivestellensatz, matrix convexity,

• etc. [Helton, McCullough, Putinar, ...].

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

(b) NC formal power series. Let Fd denote the free semigroup with d generators g1, . . . , gd (the letters) and unity ∅ (the empty word). For a word w = gj1 · · · gjm ∈ Fd its length is |w| = m ∈ N, and |∅| = 0. Let z1, . . . , zd be noncommuting indeterminates and w = gj1 · · · gjm ∈ Fd. Set zw = zj1 · · · zjm, z∅ = 1. For a linear space L, the formal power series (FPSs) in z1, . . . , zd with coefficients in L has the form f (z) =

• w∈Fd

fwzw. NC polynomials are finite FPSs: p(z) =

• w∈Fd: |w|≤m

pwzw.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results The definition Examples and motivation

One can evaluate FPSs on d-tuples of bounded linear operators (or

• f n × n matrices, n = 1, 2 . . .):

f (X) =

• w∈Fd

fw ⊗ X w (of course, provided this series converges in certain topology). NC FPSs appear in NC algebra, finite automata and formal languages, enumeration combinatorics, probability, system theory... (c) Quasideterminants, nc symmetric functions [Gelfand–Retakh...]. (d) NC continued fractions [Wedderburn]. (e) Formal Baker–Campbell–Hausdorff series [Dynkin]. (f) Analytic functions of several noncommuting variables. [J. L. Taylor] Some applications in free probability [Voiculescu].

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

Theorem

Let f : Ω → Wnc be a nc function on a nc set Ω. Let X ∈ Ωn, Y ∈ Ωm, and Z ∈ Vn×m be such that X Z

0 Y

• ∈ Ωn+m. Then

f X Z Y

• =

f (X) ∆Rf (X, Y )(Z) f (Y )

• ,

where the off-diagonal block entry ∆Rf (X, Y )(Z) is determined uniquely and is linear in Z. ∆R plays a role of a right difference-differential operator. Thus, the formula of finite differences holds: f (X) − f (Y ) = ∆Rf (Y , X)(X − Y ) n ∈ N, X, Y ∈ Ωn. The linear mapping ∆f (Y , Y )(·) plays the role of a nc differential. If K = R or C, setting X = Y + tZ with t ∈ R (t ∈ C), we obtain f (Y + tZ) − f (Y ) = t∆Rf (Y , Y + tZ)(Z).

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

Under appropriate continuity conditions, it follows that ∆Rf (Y , Y )(Z) is the directional derivative of f at Y in the direction Z. In the case of V = Kd, the finite difference formula turns into f (X) − f (Y ) =

N

• i=1

∆R,if (Y , X)(Xi − Yi), X, Y ∈ Ωn, with the right partial difference-differential operators ∆R,i: ∆R,if (Y , X)(C) := ∆Rf (Y , X)(0, . . . , 0, C

• ith place

, 0, . . . , 0). The linear mapping ∆R,if (Y , Y )(·) plays the role of a right nc i-th partial differential at the point Y . The left nc full and partial difference-differential operators ∆L, ∆L,i, i = 1, . . . , d, are defined analogously.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

Properties of ∆R (the calculus rules).

• 1. If c ∈ W and f (X) = c ⊗ In, then ∆Rf (X, Y )(Z) = 0.
• 2. ∆R(af + bg) = a∆Rf + b∆Rg for any a, b ∈ K.
• 3. If ℓ: V → W is linear, then it can be extended to

ℓ: Vn×m → Wn×m by ℓ([vij]) = [ℓ(vij)]. Then ∆Rℓ(X, Y )(Z) = ℓ(Z). In particular, if ℓj : Kd

nc → Knc is the j-th coordinate nc

function, i.e., ℓj(X) = ℓj(X1, . . . , Xd) = Xj, then ∆Rℓj(X, Y )(Z) = Zj.

• 4. If f : Ω → Xnc, g : Ω → Ync be nc functions. Assume that the

product (x, y) → x · y on X × Y with values in a vector space W over K is well defined. We extend the product to matrices

• ver X and over Y of appropriate sizes. Then

∆R(f ·g)(X, Y )(Z) = f (X)·∆Rg(X, Y )(Z)+∆Rf (X, Y )(Z)·g(Y ).

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

• 5. Let f : Ω → Anc be a nc function, where A is a unital algebra
• ver K. Let

Ωinv :=

∞

• n=1

{X ∈ Ωn : f (X) is invertible in An×n}. Then Ωinv is a nc set, f −1 : Ωinv → Anc defined by f −1(X) := f (X)−1 is a nc function, and ∆Rf −1(X, Y )(Z) = −f (X)−1∆Rf (X, Y )(Z)f (Y )−1.

• 6. Let f : Ω → Xnc and g : Λ → Ync be nc functions on nc sets

Ω ⊆ Vnc and Λ ⊆ Xnc, such that f (Ω) ⊆ Λ. Then the composition g ◦ f : Ω → Ync is a nc function, and ∆R(g ◦ f )(X, Y )(Z) = ∆Rg(f (X), f (Y ))(∆Rf (X, Y )(Z)).

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

As a function of X and Y , ∆Rf (X, Y )(·) respects direct sums and similarities, or equivalently, respects intertwinings: if X ∈ Ωn, Y ∈ Ωm, X ∈ Ω˜

n,

Y ∈ Ω ˜

m, and T ∈ K˜ n×n, S ∈ Km× ˜ m are such

that TX = XT, YS = S Y , then T∆Rf (X, Y )(Z)S = ∆Rf ( X, Y )(TZS). We denote the class of functions h on Ω × Ω whose values on Ωn × Ωm are linear mappings Vn×m → Wn×m satisfying the property above (with ∆Rf replaced by h) as T 1 = T 1(Ω; Wnc, Vnc). Thus for a nc function f , ∆Rf ∈ T 1.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

More generally, we define the class of nc functions of order k, T k = T k(Ω; W0,nc, W1,nc, . . . , Wk,nc) as a class of functions on Ωk+1, where Ω ⊆ Vnc is a nc set, whose values on Ωn0 × · · · × Ωnk are k-linear forms W1n0×n1 × · · · × Wknk−1×nk → W0n0×nk, and which respect direct sums and similarities, or equivalently, respect intertwinings... The class T 0 = T 0(Ω; Wnc) is the class of nc functions f : Ω → Wnc. We define ∆R : T k → T k+1 so that iterations ∆ℓ

R : T k → T k+ℓ are well defined...

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

Theorem

Let f ∈ T 0(Ω; Wnc). Then ∆ℓ

Rf (X 0, . . . , X ℓ)(Z 1, . . . , Z ℓ)

= f                   X 0 Z 1 · · · X 1 ... ... . . . . . . ... ... ... . . . ... X ℓ−1 Z ℓ · · · · · · X ℓ                  

1,ℓ+1

.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

We use the calculus of higher order nc difference-differential

• perators to derive a nc analogue of the Brook Taylor expansion,

which we call the Taylor–Taylor (TT) expansion in honour of Brook Taylor and of Joseph L. Taylor.

Theorem

Let f ∈ T 0(Ω; Wnc) with Ω ⊆ Vnc a nc set, n ∈ N, and Y ∈ Ωn. Then for each N ∈ N and arbitrary X ∈ Ωn, f (X) =

N

• ℓ=0

∆ℓ

Rf (Y , . . . , Y

• ℓ+1 times

)(X − Y , . . . , X − Y

• ℓ times

) + ∆N+1

R

f (Y , . . . , Y

• N+1 times

, X)(X − Y , . . . , X − Y

• N+1 times

).

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results NC difference-differential operators Higher order nc functions The Taylor–Taylor formula

In the case where V = Kd, we obtain f (X) =

N

• ℓ=0
• w=gi1···giℓ

∆w⊤

R f (Y , . . . , Y

• ℓ+1 times

)(Xi1 − Yi1, . . . , Xiℓ − Yiℓ) +

• w=gi1···giN+1

∆w⊤

R f (Y , . . . , Y

• N+1 times

, X)(Xi1 − Yi1, . . . , XiN+1 − YiN+1), where for a word w = gi1 · · · giℓ, ∆w⊤

R

:= ∆R,iℓ · · · ∆R,i1. If Y = (µ1In, . . . , µdIn), this is a genuine nc power expansion f (X) =

N

• ℓ=0
• |w|=ℓ

(X − Inµ)w ∆w⊤

R f (µ, . . . , µ

• ℓ+1 times

) +

• |w|=N+1

(X − Inµ)w ∆w⊤

R f ( µ, . . . , µ

• N+1 times

, X).

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Theorem

Let f be a nc function on Kd

nc, where K is a field of characteristic

zero, with values in Wnc (so that W is a vector space over K). Assume that for each n, f (X1, . . . , Xd) is a polynomial in dn2 commuting variables (Xi)jk, i = 1, . . . , d; j, k = 1, . . . , n, with values in Wn×n. Assume also that the degrees of these polynomials of the commuting variables (Xi)jk are bounded, (i.e., a degree bound is independent of n). Then f is a nc polynomial with coefficients in W.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Recall that a Banach space W over C is called an operator space if a sequence of norms · n on Wn×n, n = 1, 2, . . . is defined so that the following two conditions hold:

◮ For every n, m ∈ N, X ∈ Wn×n and Y ∈ Wm×m,

X ⊕ Y n+m = max{Xn, Y m}.

◮ For every n ∈ N, X ∈ Wn×n and S, T ∈ Cn×n,

SXTn ≤ S XnT, where · denotes the operator norm of Cn×n with respect to the standard Euclidean norm of Cn.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Let W be an operator space. For Y ∈ Ws×s and r > 0, define a nc ball centered at Y of radius r as Bnc(Y , r) =

∞

• m=1

B m

• α=1

Y , r

• =

∞

• m=1
• X ∈ Wms×ms :
• X −

m

• α=1

Y

• ms

< r

• .

Proposition

Let Y ∈ Ws×s and r > 0. For any X ∈ Bnc(Y , r) there is a ρ > 0 such that Bnc(X, ρ) ⊆ Bnc(Y , r). Hence, nc balls form a basis for a topology on Wnc (the uniformly-open topology). Open sets in this topology will be called uniformly open.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Let V, W be operator spaces, and let Ω ⊆ Vnc be a uniformly open nc set. A nc function f : Ω → Wnc is called uniformly locally bounded if for any s ∈ N and Y ∈ Ωs there exists a r > 0 such that Bnc(Y , r) ⊆ Ω and f is bounded on Bnc(Y , r), i.e., there is a M > 0 such that f (X)sm ≤ M for all m ∈ N and X ∈ Bnc(Y , r)sm. A nc function f : Ω → Wnc is called Gˆ ateaux (G-) differentiable if for every n ∈ N the function f |Ωn is G-differentiable, i.e., for every X ∈ Ωn and Z ∈ Vn×n the G-derivative of f at X in direction Z, δf (X)(Z) = lim

t→0

f (X + tZ) − f (X) t = d dt f (X + tZ)

• t=0,
• exists. A nc function is called uniformly analytic if f is uniformly

locally bounded and G-differentiable.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Theorem

Let a nc function f : Ω → Wnc be uniformly locally bounded. For s ∈ N, Y ∈ Ωs, let δ := sup{r > 0: f is bounded on Bnc(Y , r)}. Then f (X) =

∞

• ℓ=0

∆ℓ

Rf

     

m

• α=1

Y , . . . ,

m

• α=1

Y

• ℓ+1 times

            X −

m

• α=1

Y , . . . , X −

m

• α=1

Y

• ℓ times

      holds, with the TT series converging absolutely and uniformly, on every open nc ball Bnc(Y , r) with r < δ.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

Corollary

Let Ω ⊆ Vnc be a uniformly open nc set. Then a nc function f : Ω → Wnc is uniformly locally bounded iff f is continuous with respect to the uniformly-open topologies on Vnc and Wnc iff f is uniformly analytic.

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential calculus Algebraic and analytic results An algebraic result Analytic results

THANK YOU!

Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results