Hilbert Spaces

Identification of the operating parameters of a complex technical system important from the operational potential point of view

“…The extension of the space by a metric equation (11), scalar addition and multiplication forms normalized linear Hilbert space 20…”

Powers of Catalan generating functions for bounded operators

“…Note that $$$ {\delta}_1^{\ast n}={\delta}_n $$$ for $$$ n\in \mathrm{\mathbb{N}} $$$. This Banach algebra has identity element, $$$ {\delta}_0 $$$; its spectrum set is the closed disc $$$ \overline{D\left(0,\frac{1}{4}\right)} $$$; and its Gelfand transform is given by the $$$ Z $$$‐transform $$$$ Z(a)(z):= \sum \limits_{n=0}^{\infty }a(n){z}^n,\kern1em z\in \overline{D\left(0,\frac{1}{4}\right)}, $$$$ (Muscat [12, Example 14.35]). It is straightforward to check that $$$ Z\left({\delta}_n\right)(z)={z}^n $$$ for $$$ n\ge 0 $$$ (see, e.g., Larsen [13, pp.…”

“…The bounded operator $$$ C(T) $$$ may be considered as the image of the Catalan sequence $$$ c={\left({C}_n\right)}_{n\ge 0} $$$ in the algebra homomorphism $$$ \Phi :{\ell}^1\left({\mathrm{\mathbb{N}}}^0,\frac{1}{4^n}\right)\to \mathcal{B}(X) $$$ where $$$$ \Phi (a)x:= \sum \limits_{n\ge 0}{a}_n{T}^n(x),\kern1em a={\left({a}_n\right)}_{n\ge 0}\in {\ell}^1\left({\mathrm{\mathbb{N}}}^0,\frac{1}{4^n}\right),\kern0.30em x\in X, $$$$ that is, $$$ \Phi (c)=C(T) $$$. The $$$ \Phi $$$ algebra homomorphism (also called functional calculus) is presented in some functional analysis textbooks, for example, Muscat [12, Chapters 13 and 14].…”

“…that is, Φ(c) = C(T). The Φ algebra homomorphism (also called functional calculus) is presented in some functional analysis textbooks, for example, Muscat [12,Chapters 13 and 14].…”

Directional Kronecker algebra for -actions

“…It is known that (see [18, p. 76, Corollary 6.20]) a closed subset A of a Banach space is compact if and only if for any , there is a finite -net for A . That is, there is a finite set such that …”