Differential $\sqrt{1+B(x,x)}$ map in $C^*$-algebra

Asked Jun 21 '16 at 23:20

Active Jun 22 '16 at 00:30

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Let $A$ is $C^*$-algebra and $P \subset A$ is subset of all elements $a \in A$ such that $a > 0$ (nonnegative) and $||a|| < \frac{1}{\sqrt{1-q}}$ (norm bounded) for some $0 < q < 1$.

Let $F : P \to A$, $F(x) = \sqrt{1+B(x,x)}$ where $B : A \times A \to A$ - is sesquilinear with norm $q$, and suppose that $B$ such that $1+B(x,x)>0$ when $x \in P$. What is Frechet derivative of $F$?

edited Jun 22 '16 at 00:30

asked Jun 21 '16 at 23:20

Mykola Pochekai

1,784

Are you sure about the Existence of square root of the operator $1_A + B(x,x)$? – Zbigniew Jun 21 '16 at 23:49
Yes, suppose $1_A + B(x,x)$ is positive when $x \in P$. – Mykola Pochekai Jun 21 '16 at 23:51
Easy variant of my question is here http://math.stackexchange.com/questions/1835194/frechet-derivative-of-square-root-on-positive-elements-in-some-c-algebra – Mykola Pochekai Jun 22 '16 at 00:28
If $B$ is bilinear and not sesquilinear, I fail to see how you know that $1_A+B(x,x)$ is positive. – Martin Argerami Jun 22 '16 at 00:29
Sesquilinear also, I will fix my question. – Mykola Pochekai Jun 22 '16 at 00:30
@MartinArgerami please, look at my another topic, maybe its obvious for you :) – Mykola Pochekai Jun 22 '16 at 00:32

Differential $\sqrt{1+B(x,x)}$ map in $C^*$-algebra

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