# Exponential of positive self-adjoint operator

In a physics paper I am currently reading, the following statement seems to be used: Let $A$ be a (not necessarily bounded) positive self-adjoint operator on $L^2(a,b)$ with $C_0^{\infty}(a,b)\subseteq \mathrm{dom}(A)$. Then $\Phi(t)=\cos(A^{1/2}t)$ is well-defined for $t\in\mathbb{R}$.

Is this statment true? And if yes, how can I prove it?

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Is it $\phi(t)=\cos(A^{1/2}t)$? – Mhenni Benghorbal Mar 18 '13 at 13:54
Yes, that's better. – Scipio Mar 18 '13 at 14:02
I don't have time to give a full answer right now, but you should look into the Borel functional calculus. Not only is $\Phi(t)$ defined for all $t$, it extends to a bounded operator on $L^2(a,b)$ since $\cos$ is bounded on $\mathbb{R}$. A reference I particularly like for material on operator algebras and functional analysis is Kadison and Ringrose's "Fundamentals of the Theory of Operator Algebras." Volume I is the appropriate one for this material. – J. Loreaux Mar 18 '13 at 14:33
What you are doing is nothing but an extension of the idea of functions of matrices to bounded linear operators. Read under Cauchy Integral. – Mhenni Benghorbal Mar 18 '13 at 22:57