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-Verbovetskyi 1 Department of Mathematics Drexel University (Philadelphia, PA) October 3, 2010 Bill Helton Workshop, UCSD 1 A joint work with V. Vinnikov Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results For a vector space V over a field K , we define the noncommutative (nc) space over V ∞ � V n × n . V nc = n =1 For X ∈ V n × n and Y ∈ V m × m we define their direct sum � X � 0 ∈ V ( n + m ) × ( n + m ) . X ⊕ Y = 0 Y Notice that matrices over K act from the right and from the left on matrices over V by the standard rules of matrix multiplication: if X ∈ V p × q and T ∈ K r × p , S ∈ K q × s , then TX ∈ V r × q , XS ∈ V p × s . A subset Ω ⊆ V nc is called a nc set if it is closed under direct sums; explicitly, denoting Ω n = Ω ∩ V n × n , we have X ⊕ Y ∈ Ω n + m for all X ∈ Ω n , Y ∈ Ω m . Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results In the case of V = K d we identify matrices over V with d -tuples of matrices over K : � K d � p × q ∼ � K p × q � d . = Under this identification, for d -tuples X = ( X 1 , . . . , X d ) ∈ ( K n × n ) d and Y = ( Y 1 , . . . , Y d ) ∈ ( K m × m ) d , �� X 1 � � X d �� � K ( n + m ) × ( n + m ) � d 0 0 X ⊕ Y = ∈ ; , . . . , 0 Y 1 0 Y d and for a d -tuple X = ( X 1 , . . . , X d ) ∈ ( K p × q ) d and matrices T ∈ K r × p , S ∈ K q × s , � K r × q � d , � K p × s � d . TX = ( TX 1 , . . . , TX d ) ∈ XS = ( X 1 S , . . . , X d S ) ∈ Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results Let V and W be vector spaces over K , and let Ω ⊆ V nc be a nc set. A mapping f : Ω → W nc with f (Ω n ) ⊆ W n × 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 ∈ Ω n and S ∈ K n × n is invertible with SXS − 1 ∈ Ω n , then f ( SXS − 1 ) = Sf ( X ) S − 1 . (2) Proposition A mapping f : Ω → W nc with f (Ω n ) ⊆ W n × n respects direct sums and similarities, i.e., (1) and (2) hold iff f respects intertwinings: for any X ∈ Ω n , Y ∈ Ω m , and T ∈ K n × 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 The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results (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 , or 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 optimization 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 The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results (b) NC formal power series . Let F d denote the free semigroup with d generators g 1 , . . . , g d (the letters ) and unity ∅ (the empty word ). For a word w = g j 1 · · · g j m ∈ F d its length is | w | = m ∈ N , and |∅| = 0. Let z 1 , . . . , z d be noncommuting indeterminates and w = g j 1 · · · g j m ∈ F d . Set z w = z j 1 · · · z j m , z ∅ = 1 . For a linear space L , the formal power series (FPSs) in z 1 , . . . , z d with coefficients in L has the form � f w z w . f ( z ) = w ∈F d NC polynomials are finite FPSs: � p w z w . p ( z ) = w ∈F d : | w |≤ m Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions The definition NC difference-differential calculus Examples and motivation Algebraic and analytic results One can evaluate FPSs on d -tuples of bounded linear operators (or of n × n matrices, n = 1 , 2 . . . ): � f w ⊗ X w f ( X ) = w ∈F d (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 operators NC difference-differential calculus Higher order nc functions Algebraic and analytic results The Taylor–Taylor formula Theorem Let f : Ω → W nc be a nc function on a nc set Ω . Let X ∈ Ω n , � X Z � Y ∈ Ω m , and Z ∈ V n × m be such that ∈ Ω n + m . Then 0 Y �� X �� � f ( X ) � Z ∆ R f ( X , Y )( Z ) f = , 0 Y 0 f ( Y ) where the off-diagonal block entry ∆ R f ( 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 ) = ∆ R f ( Y , X )( X − Y ) n ∈ N , X , Y ∈ Ω n . 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 ∆ R f ( Y , Y + tZ )( Z ) . Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results

The definition and examples of nc functions NC difference-differential operators NC difference-differential calculus Higher order nc functions Algebraic and analytic results The Taylor–Taylor formula Under appropriate continuity conditions, it follows that ∆ R f ( Y , Y )( Z ) is the directional derivative of f at Y in the direction Z . In the case of V = K d , the finite difference formula turns into N � f ( X ) − f ( Y ) = ∆ R , i f ( Y , X )( X i − Y i ) , X , Y ∈ Ω n , i =1 with the right partial difference-differential operators ∆ R , i : ∆ R , i f ( Y , X )( C ) := ∆ R f ( Y , X )(0 , . . . , 0 , C , 0 , . . . , 0) . ���� i th place The linear mapping ∆ R , i f ( 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 operators NC difference-differential calculus Higher order nc functions Algebraic and analytic results The Taylor–Taylor formula Properties of ∆ R (the calculus rules). 1. If c ∈ W and f ( X ) = c ⊗ I n , then ∆ R f ( X , Y )( Z ) = 0. 2. ∆ R ( af + bg ) = a ∆ R f + b ∆ R g for any a , b ∈ K . 3. If ℓ : V → W is linear, then it can be extended to ℓ : V n × m → W n × m by ℓ ([ v ij ]) = [ ℓ ( v ij )]. Then ∆ R ℓ ( X , Y )( Z ) = ℓ ( Z ) . In particular, if ℓ j : K d nc → K nc is the j-th coordinate nc function , i.e., ℓ j ( X ) = ℓ j ( X 1 , . . . , X d ) = X j , then ∆ R ℓ j ( X , Y )( Z ) = Z j . 4. If f : Ω → X nc , g : Ω → Y nc 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 over X and over Y of appropriate sizes. Then ∆ R ( f · g )( X , Y )( Z ) = f ( X ) · ∆ R g ( X , Y )( Z )+∆ R f ( X , Y )( Z ) · g ( Y ) . Dmitry Kaliuzhnyi-Verbovetskyi Noncommutative functions: Algebraic and analytic results