Let X be a Complex Banach space and T be a bounded operator in X. The number sup {|λ| : λ ∈ σ(T)} (where σ(T) is the spectrum of T and σ(T) ̸= ϕ) is called the spectral radius of T and denoted by r(T). Since λ ≤ ∥T∥for all λ ∈ σ(T), it follows that r(T) ≤ ∥T∥. The spectral mapping theorem implies that r(Tn) = (r(T))n for every positive integer n. It frequently turns out that it is easy to compute the spectral radius of an operator even if it is hard to _nd the spectrum. This is often made easy by the spectral radius formula. Let H be a Hilbert space and T be a bounded linear operator in H. In this paper we show that if T is normal, then Tn is normal for each n ∈ N and ∥Tn∥ = ∥T∥n. Consequently, we use the spectral radius formula to show that r(T) = ∥T∥. Moreover, we show that if X is a Complex Banach space and T is bounded in X then there is a λ belonging to the spectrum of T such that |λ| = r(T). Let H be a Complex Hilbert space and T be a bounded operator in H which is normal; we show that ∥T∥ = sup {|Tx, x| : x ∈ H and ∥x∥ = 1} and the residual spectrum of T is void.

The concept of a compact operator on a Hilbert space, H is an extension of the concept of a matrix acting on a finite-dimensional vector space. In Hilbert space, compact operators are precisely the closure of finite rank operators in the topology induced by the operator norm. In this paper, we provide an elementary exposition of compact linear operators in pre-Hilbert and Hilbert spaces. However, whenever advantageous, we may prove a few results in the general context of normed linear spaces. It is well known that strong convergence implies weak convergence but weak convergence does not imply strong convergence. We also show that an operator T \(\epsilon\) B(H) is compact if and only if T maps every weakly convergent sequence in H to a strongly convergent sequence.

It is possible to express all geometric notions connected with closed linear subspaces in terms of algebraic properties of the orthoprojectors onto these linear spaces. In this paper, sucient conditions for the calculus of a family of orthoprojectors in B(H) have been given with meaningful consideration of the sum, the product and dierence of orthoprojectors to be a projector. This has been done by giving the algebraic formulations of orthogonality for the sum, product and dierence. From the paper, it is observed that there is a natural one-to-one correspondence between the set of all closed linear subspaces of a Hilbert space H and the set of all orthoprojectors on H. This paper will help in the study of vector space with many diverse applications such as orthogonal polynomials, QR decomposition of projectors and Gram-Schmidt orthogonalization.

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.