EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5587 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Closability of Class Totally Paranormal Operators Salam Alnabulsi1, M.H.M. Rashid2,∗ 1 Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan 2 Department of Mathematics, Faculty of Science P.O.Box 7, Mutah University, Al-Karak, Jordan Abstract. This article explores the analysis of various spectral properties pertaining to totally paranormal closed operators, extending beyond the confines of boundedness and encompassing operators defined in a Hilbert space. Within this class, closed symmetric operators are included. Initially, we establish that the spectrum of such an operator is non-empty and provide a char- acterization of closed-range operators in terms of the spectrum. Building on these findings, we proceed to prove Weyl’s theorem, demonstrating that for a densely defined closed totally paranor- mal operator T , the difference between the spectrum σ(T ) and the Weyl spectrum σw(T ) equals the set of all isolated eigenvalues with finite multiplicities, denoted by π00(T ). In the final section, we establish the self-adjointness of the Riesz projection Eµ corresponding to any non-zero isolated spectral value µ of T . Furthermore, we show that this Riesz projection satisfies the relationships ran(Eµ) = ker(T − µI) = ker(T − µI)∗. Additionally, we demonstrate that if T is a closed totally paranormal operator with a Weyl spectrum σw(T ) = 0, then T qualifies as a compact normal operator. 2020 Mathematics Subject Classifications: 47A10,47A53, 47B20 Key Words and Phrases: Densely defined operator, closed operator, totally paranormal, reduced minimum modulus, Riesz projection, Weyl’s theorem 1. Introduction The class of normal operators is fundamental in operator theory, having been the sub- ject of significant research. The spectral theorem for these operators confirms the existence of non-trivial invariant subspaces and provides insight into the operator’s full structure. The category of bounded paranormal operators was initially investigated by Istrǎtescu, who referred to it as class N [13]. Later, Furuta coined the term ”paranormal operator” [8]. Numerous researchers have since studied bounded paranormal operators, including ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5587 Email addresses: s.alnabulsi@ju.edu.jo (S. Alnabulsi), malik okasha@yahoo.com (M.H.M. Rashid) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 2 of 20 works by Ando and others [3, 8, 13]. Specifically, Ando [3] provided a characterization of bounded paranormal operators, while Istrǎtescu demonstrated that normaloid operators generalize paranormal operators [13]. A continuous linear operator on a complex Banach space is said to be paranormal if ∥Tx∥2 ≤ ∥∥T 2x ∥∥ ∥x∥ for all x ∈ X, where X is a Banach space. T is called totally paranormal [22] if T − µI is paranormal for every µ ∈ C. That is, ∥(T − µI)x∥2 ≤∥∥(T − µI)2x ∥∥ ∥x∥ for all x ∈ X and µ ∈ C. Hence, We have the following inclusion relation between some subclasses and a generalized class of bounded totally paranormal operators. Normal ⊆ Hyponormal ⊆ Totally Paranormal ⊆ Paranormal ⊆ normaloid. The inclusion relationships mentioned above are strict. For additional information, see [8, 22]. Daniluk extended the concept of bounded paranormal operators to encompass unbounded operators, exploring the conditions for their closability [6]. In this paper, we focus on densely defined, closed totally paranormal operators in a Hilbert space H and establish the following results. Let T be a densely defined closed totally paranormal operator in H. Then (i) spectrum of T is non-empty. (ii) Every isolated spectral value of T is an eigenvalue. (iii) In addition, if ker(T ) = ker(T ∗), then (a) range of T is closed if and only if 0 is an isolated spectral value of T . (b) The minimum modulus, m(T ) is equal to the distance of 0 from spectrum of T . (iv) T satisfies the Weyl’s Theorem i.e. σ(T )\σw(T ) = π00(T ). Here σw(T ) is the Weyl’s spectrum and π00(T ) consists of all isolated eigenvalues of T with finite multiplicity. (v) If µ is a non-zero isolated spectral value of T , then the Riesz projection Eµ with respect to µ is self-adjoint and satisfies ran(Eµ) = ker(T − µI) = ker(T − µI)∗. The study of Weyl’s theorem and the self-adjointness of the Riesz projection for isolated spectral values has been explored for various operator classes. Coburn [5] established these properties for certain non-normal operators, including hyponormal and Toeplitz operators. Schmoeger [22] expanded this work to bounded totally paranormal operators, drawing on Ando’s characterization [3] of paranormal operators. Since Ando’s characterization does not extend to unbounded paranormal operators, and methods for bounded operators are unsuitable here, we aim to establish properties (iv) and (v) using an alternative approach. Gupta and Mamtani [11] showed that closed hyponormal operators satisfy Weyl’s the- orem. In a follow-up study [10], they outlined key conditions required for the orthogonal direct sum of densely defined closed operators to fulfill Weyl’s theorem. The paper is organized into four sections for clarity. In Section 2, we introduce key no- tations and summarize relevant established results that will be used throughout the study. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 3 of 20 Section 3 focuses on examining spectral properties associated with densely defined closed totally paranormal operators. Finally, Section 4 presents the proof of Weyl’s theorem for these operators. 2. Notations and preliminaries In this article, we explore intricate Hilbert spaces, represented as H,H1,H2, and so forth. The inner product and the corresponding norm are symbolized by ⟨·, ·⟩ and ∥·∥, respectively. The set of all linear operators on H is denoted as L(H), while the collection of all bounded linear operators is represented as B(H). For a linear operator T ∈ L(H), we use D(T ), ker(T ), and ran(T ) to signify its domain, null space, and range space, respectively. A linear operator T is termed a densely defined operator if D(T ) = H. If T ∈ L(H) and M is a closed subspace of H , then M is said to be invariant under T , if for every x ∈ D(T )∩M, Tx is in M. We denote the identity operator on M by IM, the orthogonal projection on M by PM. The unit sphere of M is TM := {x ∈ M : ∥x∥ = 1}. The restriction of T to M is an operator T |M : M ∩D(T ) → H defined by T |Mx = Tx, for all x ∈ M∩D(T ). If M is invariant under T , then T |M is an operator from D(T )∩M into M. An operator T ∈ L(H) is said to be closed if for any sequence {xn} ⊆ D(T ) with xn → x and Txn → y then x ∈ D(T ) and Tx = y. In this document, the notation C(H1,H2) will be employed to denote the collection of closed linear operators such that D(T ) ⊆ H1 and ran(T ) ⊆ H2. In the case where H1 equals H2, we will use the shorthand C(H). It is known that every densely defined operator T ∈ C(H1,H2) has a unique adjoint in C(H2,H1), that is, there exists a unique T ∗ ∈ C(H2,H1) such that ⟨Tx, y⟩ = ⟨x, T ∗y⟩ for all x ∈ D(T ) and y ∈ D(T ∗). Remark 1. By the closed graph theorem (cf. [18, Theorem 7.1, page 231], it follows that a closed operator T ∈ C(H1,H2) with D(T ) = H1 is bounded. Lemma 1. [9] Let T ∈ L(H) be a densely defined closed operator. Then D(T ) ∩ ker(T )⊥ = ker(T )⊥. If S and T are two closed operators, then S is called an extension of T (or T is a restriction of S), if D(T ) ⊆ D(S) and Sx = Tx for all x ∈ D(T ). This is often denoted as T ⊆ S. Consequently, S = T if and only if D(S) = D(T ) and Sx = Tx for all x ∈ D(S) = D(T ). Definition 1. [25] A densely defined operator T ∈ L(H) is said to be self-adjoint operator if D(T ) = D(T ∗) and T = T ∗. And a self-adjoint operator T is said to be positive if ⟨Tx, x⟩ ≥ 0 for all x ∈ D(T ). S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 4 of 20 Definition 2. [20, Page 365] If T ∈ L(H) is a closed operator, then the resolvent set of T is defined by ρ(T ) = { µ ∈ C : T − µI is invertible and (T − µI)−1 ∈ B(H) } and the spectrum of T , denoted by σ(T ), is defined by σ(T ) := C \ ρ(T ) Note that σ(T ) is a closed subset of C. Moreover σ(T ) can be empty set or the whole complex plane C. The spectrum of T decomposes as the disjoint union of the point spectrum σp(T ), the continuous spectrum σc(T ) and the residual spectrum σr(T ), where σp(T ) = {µ ∈ C : T − µI is not injective} , σr(T ) = {µ ∈ C : T − µI is not injective but ran(T − µI) is not dense in H} , σc(T ) = σ(T ) \ (σp(T ) ∪ σr(T )) . The spectral radius of T ∈ B(H) is defined by r(T ) := sup {|µ| : µ ∈ σ(T )} . An operator T ∈ B(H) is said to be normaloid, if r(T ) = ∥T∥. Recall that a linear operator T ∈ L(H) is compact, if T maps every bounded set in H to a pre-compact set in H. For more details about compact operators, we refer to [21]. Definition 3. [21, Page 156] A closed operator T in a densely defined space H is termed Fredholm if ran(T ) is closed, and both the dimensions of ker(T ) and its orthogonal comple- ment ran(T )⊥ are finite. In such instances, the index of T , denoted by ind(T ), is defined as ind(T ) = dim(ker(T ))− dim(ran(T )⊥). Remark 2. If T ∈ L(H) is a densely defined closed Fredholm operator and K is a compact operator, then T +K is also Fredholm and ind(T +K) = ind(T ). Definition 4. [21, Page 172] If T ∈ L(H) is a densely defined closed operator, then the Weyl’s spectrum of T is defined by σw(T ) = {λ ∈ C : T − λI is not Fredholm of index 0} and π00(T ) = {λ ∈ σp(T ) : λ is isolated with dim (ker(T − λI)) < +∞}. Suppose T ∈ L(H) is a densely defined closed operator with σ(T ) = σ ∪ τ , where σ is contained in some bounded domain ∆ and τ is a subset of the complement of ∆. Let Λ be the boundary of ∆, then Eσ = 1 2πi ∫ Λ (zI − T )−1dz (1) is called the Riesz projection with respect to σ. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 5 of 20 Theorem 1. [12, Theorem 2.1, Page 326] Suppose T ∈ L(H) is a densely defined closed operator with σ(T ) = σ ∪ τ , where σ is contained in some bounded domain and Eσ is the operator defined in Equation (1). Then (i) Eσ is a projection. (ii) The subspaces ran(Eσ) and ker(Eσ) are invariant under T . (iii) The subspace ran(Eσ) is contained in D(T ) and T |ran(Eσ) is bounded. (iv) σ ( T |ran(Eσ) ) = σ and σ(T |ker(Eσ)) = τ . In particular, if µ is an isolated point of σ(T ), then there exist a positive real number r such that {z ∈ C : |z−µ| ≤ r} ∩ σ(T ) = {µ}. If we take Λ = {z ∈ C : |z − µ| = r}, then the Riesz projection with respect to µ is defined by Eµ = 1 2πi ∫ Λ (zI − T )−1dz. (2) Definition 5. [16] Let T ∈ L(H) be a closed operator. Then (i) the minimum modulus of T is defined by m(T ) := inf { ∥Tx∥ : x ∈ TD(T ) } . Then (ii) the reduced minimum modulus of T isdenoted by γ(T ) := inf { ∥Tx∥ : x ∈ TD(T )∩ker(T )⊥ } . By the definition, it is clear that m(T ) ≤ γ(T ) The following characterization of closed range operators is frequently used in the article. Theorem 2. [1, Page 334] For a densely defined closed operator T ∈ L(H), the following are equivalent. (i) ran(T ) is closed. (ii) ran(T ∗) is closed. (iii) γ(T ) > 0. (iv) S0 = T |D(T )∩ker(T )⊥ has a bounded inverse. If T ∈ L(H) is a densely defined closed operator and ker(T ) = {0}, then the inverse operator, T−1 is the linear operator from H to H, with D(T−1) = ran(T ) and T−1Tx = x for all x ∈ D(T ). In particular if T is a bijection, then by the closed graph theorem it follows that T−1 ∈ B(H). In addition, if T is normal then T has a bounded inverse if and only if m(T ) > 0. Theorem 3. If T ∈ B(H) is a totally paranormal, then (i) T is normaloid. (ii) T−1 is totally paranormal, if T is invertible. (iii) T is unitary, if σ(T ) lies on the unit circle. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 6 of 20 3. Spectral Properties In this section, we study some spectral properties of densely defined closed totally paranormal operators. Definition 6. A densely defined operator T ∈ L(H) is said to be totally paranormal if ∥(T − λI)x∥2 ≤ ∥∥(T − λI)2x ∥∥ ∥x∥ for all λ ∈ C and D((T − λ)2) ⊆ D(T − λ). Equivalently, T is totally paranormal if and only if T − λI is paranormal for all λ ∈ C. And T is totally ∗-paranormal if T − λI is ∗-paranormal for all λ ∈ C. A densely defined operator T in H is said to be hyponormal if D(T ) ⊆ D(T ∗) and ∥T ∗x∥ ≤ ∥Tx∥ for x ∈ D(T ). And T is cohyponormal if T ∗ is hyponormal. Proposition 1. Let T ∈ C(H). Then (i) If T is closed cohyponormal and closed totally ∗-paranormal, then T is closed totally paranormal. (ii) If T is closed hyponormal and closed totally paranormal, then T is closed totally ∗-paranormal. Proof. (i) Assume that T is closed cohyponormal and closed totally ∗-paranormal. If T is closed cohyponormal, then so is T − λI for all λ ∈ C. Hence, D((T − λI)2) ⊆ D((T − λI)∗) ⊆ D(T − λI). Let x ∈ D((T − λI)2). Then we have ∥(T − λI)x∥2 ≤ ∥(T − λI)∗x∥2 ≤ ∥∥(T − λI)2x ∥∥ ∥x∥ . (ii) By the hypotheses on T , D((T − λI)2) ⊆ D(T − λI) ⊆ D((T − λI)∗), and for each x ∈ D((T − λI)2), ∥(T − λI)∗x∥2 ≤ ∥(T − λI)x∥2 ≤ ∥∥(T − λI)2x ∥∥ ∥x∥ . So, the proof is complete. Proposition 2. Let T ∈ C(H) be totally ∗-paranormal, and let λ be any complex scalar. Then ker(T − λI) ⊂ ker(T − λI)∗. Proof. Let x ∈ D(T ) be a unit eigenvector of T associated to λ. Then Tx = λx. By the hypotheses on T , ∥T ∗x∥ ≤ |λ|. Consequently, 0 ≤ ∥∥T ∗x− λ̄x ∥∥2 = ∥T ∗x∥2 − λ ⟨x, Tx⟩ − λ̄ ⟨Tx, x⟩+ |λ|2 = ∥T ∗x∥2 − |λ|2 ≤ 0. Thus, T ∗x = λ̄x and so x ∈ ker(T − λ)∗. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 7 of 20 Lemma 2. Let T ∈ C(H) be a totally ∗-paranormal, then there exists a contraction Qλ ∈ B(H) such that (T − λI)2 ⊂ (T − λI)∗Qλ. Proof. Since ∥(T − λI)∗x∥2 ≤ ∥∥(T − λI)2x ∥∥ ∥x∥ for all x ∈ D((T − λI)2) ⊆ D((T − λI)∗), there exists a contraction K ′ ∈ B ( ran ((T − λI)2), ran(T ∗ − λ̄I) ) such that K ′(T− λI)2 ⊂ T ∗ − λ̄I. Let K ∈ B(H) be any contraction which extends K ′ (e.g. set Kx = 0 for x ∈ H⊖ran ((T − λI)2)). ThenK(T−λI)2 ⊂ T ∗−λ̄I. Taking adjoints in the last inclusion and exploiting the closability of T , we get (T −λI)2 ⊆ ( (T − λI)2 )∗∗ ⊆ (K(T − λI)2 )∗ = (T − λI)∗2K∗ ⊆ (T − λI)∗K∗. This gives us the conclusion with Q = K∗. A closed subspace M of H reduces T ∈ C(H) if M and M⊥ are invariant under T . Stochel [23] proved if T ∈ C(H) is hyponormal and M is a closed subspace of H which is invariant under T with T |M is normal, then M reduces T . Theorem 4. Suppose T ∈ C(H) is a densely defined totally ∗-paranormal. If M is a closed subspace of H which is invariant under T with T |M is normal, then M reduces T . Proof. Let H = H1⊕H2, where H1 = M and H2 = M⊥. Then T has the block matrix representation [ T11 T12 T21 T22 ] , where Tij : D(T ) ∩ Hj → Hi is defined by Tij = P |HiTP |Hj |D(T )∩Hk for k = 1, 2. Here, P |Hi denotes the orthogonal projection onto Hi. Since M is invariant under T , we have[ T11 T12 0 T22 ] . Let y ∈ D(T ) ∩M⊥. By Lemma 2, we have (T − λI)2 ⊂ (T − λI)∗Qλ for every λ ∈ C. Thus, ran ( (T − λI)2 ) ⊆ ran ((T − λI)∗)for every λ ∈ C. Then there exist a densely defined operator B such that (T − λI)2 = (T − λI)∗B (see [7]). Hence, T12(y) = (T11 − λI)∗u for some u ∈ M. We can choose v such that (T11 − λI)∗u = (T11 − λI)v. Therefore, T12(y) = (T11 − λI)v for every λ ∈ C. Consequently, T12(y) ∈ ⋂ λ∈C ran(T11 − λI). Hence, T12(y) = 0 for all y ∈ D(T ) ∩M⊥ (see [19]) and so T12 = 0. This ends the proof. The ascent p(T ) and descent q(T ) of an operator C(H) are given by p(T ) = inf { n : ker(Tn) = ker(Tn+1) } and q(T ) = inf { n : ran(Tn) = ran(Tn+1) } It follows from the definition of totally paranormal the following result holds. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 8 of 20 Proposition 3. Let T ∈ L(H) be a densely defined closed totally paranormal operator. Then the ascent p(T ) and descent q(T ) of T are finite for all λ ∈ C. Definition 7. [14] Let T be a non necessarily bounded operator with domain D(T ) ⊂ H. We say that λ is not in σ(T ) if T − λ is injective and (T − λ)−1 ∈ B(H). The following results immediately follows from the definition. Proposition 4. Let T ∈ L(H) be a densely defined closed totally paranormal operator. Then T − αI and αT are totally paranormal operators for all α ∈ C. Proposition 5. Let Let T ∈ L(H) be a densely defined closed totally paranormal operator. If σ(T ) = {µ}, then T = µI. We now give the counterexample of a closed densely defined operator T such that both T and T ∗ are one-to-one and totally paranormal, yet T is not normal. Example 1. The Hilbert space in question is L2(R)⊕L2(R). From [24], we have an explicit example of a densely defined unbounded closed operator T for which D(T 2) = D(T ∗2) = {0} More precisely, T is defined by T = [ 0 A−1 B 0 ] on D(T ) := D(B) ⊕ D(A−1) ⊂ L2(R) ⊕ L2(R), and where A and B are two unbounded self-adjoint operators such that D(A) ∩D(B) = D(A−1) ∩D(B−1) = {0}, where A−1 and B−1 are not bounded (as in [15]). Hence T ∗ = [ 0 B A−1 0 ] for A−1 and B are both self-adjoint. Observe now that both T and T ∗ are one-to-one since both A−1 and B are so. Both T and T ∗ are trivially totally paranormal thanks to the assumption D(T 2) = D(T ∗2) = {0}. So tatally paranormality of both operators need only be checked at the zero vector and this is plain as ∥(T − µI)x∥2 = ∥∥(T − µI)2x ∥∥ ∥x∥ = 0 and ∥(T − µI)∗x∥2 = ∥∥(T − µI)2x ∥∥ ∥x∥ = 0 for x = 0. However, T cannot be normal for it were, T 2 would be normal too, in particular it would be densely defined which is impossible here Here we discuss some basic results related to unbounded totally paranormal operators, which are often used in the article. Theorem 5. Let T ∈ L(H) be a densely defined closed totally paranormal operator and not a multiple of the identity. Then the following holds. (i) If M is a closed invariant subspace of T , then T |M is totally paranormal. (ii) If 0 /∈ σ(T ), then T−1 totally paranormal S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 9 of 20 (iii) σ(T ) is nonempty. Proof. (i) For every λ ∈ C, as M is invariant under T , we have D((T − λI)2|M) = D ( (T − λI)2 ) ∩M = {x ∈ D(T − λI) : (T − λI)x ∈ D(T − λI)} ∩M = {x ∈ D(T − λI) ∩M : (T − λI)x ∈ D(T − λI) ∩M} (since T (D(T − λI) ∩M) ⊆ M) = {x ∈ D((T − λI)|M) : Tx ∈ D((T − λI)|M)} = D ( ((T − λI)|M)2 ) . Thus, (T − λI)2|M = ((T − λ)|M)2. Now the result follows from the below inequality; ∥(T − λI)|Mx∥2 = ∥(T − λI)x∥2 ≤ ∥∥(T − λI)2x ∥∥ = ∥∥(T − λI)2|Mx ∥∥ = ∥∥∥((T − λI)|M)2 x ∥∥∥ , ∀x ∈ TD(((T−λI)|M)2). (ii) Existence of T−1 implies ran(T ) = H and consequently ran ( (T )2 ) = H. As T is totally paranormal, we get ker(T ) = ker((T )2), so T 2 is bijective and ( (T )2 )−1 exists. Also D( ( (T )−1 )2 ) = H = ran ( (T )2 ) . If y ∈ H, then there exist x ∈ D ( (T )2 ) , such that y = T 2x. Now, ∥∥T−1y ∥∥2 = ∥Tx∥2 ≤ ∥∥T 2x ∥∥ ∥x∥ = ∥y∥ ∥∥T−2y ∥∥ . Hence T−1 is totally paranormal since totally paranormal has invariant translation prop- erty. (iii) Suppose on the contrary that σ(T ) = ∅. Then T is invertible and T ∈ B(H). First, we show that σ(T−1) = {0}. For any complex number µ ̸= 0, consider the operator S = µ−1(T − µ−1I)−1. Here S can also be written as the sum of two bounded opera- tors, S = µ−1(I + µ−1(T − µ−1I)−1), so S is bounded. By a simple computation we can show that S is the bounded inverse of µI − T−1. Thus σ(T−1) ⊆ {0}. As T−1 ∈ B(H), this implies σ(T−1) is non-empty, so we conclude that σ(T−1) = {0}. By (ii), T−1 is bounded totally paranormal operator and consequently normaloid by Theorem 3. Hence∥∥T−1 ∥∥ = 0, which implies T−1 = 0, a contradiction. Hence σ(T ) is non-empty. Now we discuss about isolated spectral values of totally paranormal operators. Theorem 6. Let T be a densely defined closed totally paranormal operator. If µ is an isolated point of σ(T ), then ker(T − µI) = ran(Eµ). Proof. It follows from [4, Lemma 3.4] that ker(T − µI) ⊆ ran(Eµ). To complete the proof we have to show that ker(T − µI) ⊇ ran(Eµ). As a consequence of Theorem 1 and Theorem 5, we know that T |ran(Eµ) is bounded S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 10 of 20 and totally paranormal. By Theorem 3 it follows that T |ran(Eµ) is normaloid. If µ = 0, then σ ( T |ran(E0) ) = {0}. This implies ∥∥T |ran(E0) ∥∥ = 0 and consequently T |ran(E0) = 0. Hence ran(E0) ⊆ ker(T ). If µ ̸= 0, then σ ( µ−1T |ran(Eµ) ) = {1}. By Theorem 3, it follows that µ−1T |ran(Eµ) is unitary. Thus T |ran(Eµ) − µIran(Eµ) is normal and σ ( T |ran(Eµ) − µIran(Eµ) ) = {0}. Since every normal operator is normaloid, we conclude that T |ran(Eµ) − µIran(Eµ) = 0. Hence ran(Eµ) ⊆ ker (T − µI). Theorem 7. Let T ∈ L(H) be a densely defined closed totally paranormal operator and µ be an isolated point of σ(T ). Then ker(Eµ) = ran(T − µI). Proof. By Theorem 1, µ /∈ σ ( T |ker(Eµ) ) . This implies that ran(T −µI)|ker(Eµ) = n(Eµ) and consequently n(Eµ) ⊆ ran(T − µI). Let y ∈ ran(T − µI). There exist x ∈ D(T ) such that y = (T − µI)x. Since H = ran(Eµ) + ker(Eµ) and ran(Eµ) ∩ ker(Eµ) = {0}, we have x = p + q, where p ∈ ran(Eµ) and q ∈ ker(Eµ). It follows from Theorem 6, that p ∈ ker(T − µI) ⊆ D(T ) and consequently q = x − p ∈ D(T ). As we know from Theorem 1 that ker(Eµ) is invariant under T , we have y = (T − µI)x = (T − µI)q ∈ (T − µI)(ker(Eµ)) ⊆ ker(Eµ). Hence ran(T − µI) ⊆ ker(Eµ). This proves the result. The following results are consequences of Theorem 7 which gives a characterization for closed range totally paranormal operators. Corollary 1. Suppose T ∈ L(H) is a densely defined closed totally paranormal operator. If 0 is an isolated point of σ(T ), then ran(T ) is closed. In general, the converse of Corollary 1 is not true. We have the following example to illustrate this. Example 2. Let T : ℓ2(N) → ℓ2(N) be defined by T (x1, x2, · · · ) = (0, x1, x2, · · · ), for all (xn) ∈ ℓ2(N) Then σ(T ) = {z ∈ C : |z| ≤ 1}, ran(T ) = ℓ2(N) \ span {e1}. Here ran(T ) is closed but 0 is not an isolated point of σ(T ). Clearly, T is a totally paranormal operator. Next result gives a sufficient condition under which the converse of Corollary 1 is also true. Theorem 8. Let T ∈ L(H) be a densely defined closed totally paranormal operator with ker(T ) = ker(T ∗) and 0 ∈ σ(T ). Then 0 is an isolated point of σ(T ) if and only if ran(T ) is closed. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 11 of 20 Proof. The necessary condition follows from Corollary 1. To prove the sufficient condi- tion. Assume that ran(T ) is closed. Consider S0 = T |ker(T )⊥ : ker(T )⊥∩D(T ) → ker(T )⊥. Clearly S0 is injective and ran(S0) = ran(T ) is closed. Also ran(S0) = ker(T ∗)⊥ = ker(T )⊥, consequently S0 is bijective and S−1 0 ∈ B(ker(T )⊥). Thus 0 /∈ σ(S0). Applying [2, Theo- rem 5.4, Page 289], σ(T ) ⊆ {0} ∪ σ(S0). Since 0 ∈ σ(T ), we have σ(T ) = {0} ∪ σ(S0) and hence 0 is an isolated point of σ(T ). Note that Theorem 8 does not hold if we drop the condition ker(T ) = ker(T ∗). Consider the operator T defined in Example 2. Clearly ker(T ) = {0} ̸= span{e1} = ker(T ∗), and ran(T ) is closed but 0 is not an isolated point of σ(T ). Theorem 9. Let T ∈ L(H) be a densely defined closed totally paranormal operator. If ker(T ) = ker(T ∗), then m(T ) = d(0, σ(T )), the distance between 0 and σ(T ). Proof. We will prove this result by considering the following two cases, which exhaust all the possibilities. Case (1): T is not injective. Clearly m(T ) = 0 and 0 ∈ σp(T ). Hence m(T ) = 0 = d(0, σ(T )). Case (2): T is injective. It suffices to show that γ(T ) = d(0, σ(T )) because m(T ) = γ(T ). First assume that γ(T ) = 0. It follows from Theorem 2 that ran(T ) is not closed and consequently 0 ∈ σc(T ). Thus d(0, σ(T )) = 0 = γ(T ). Now assume that γ(T ) > 0. As a consequence of Theorem 2, ran(T ) is closed. Note that 0 /∈ σ(T ), otherwise Theorem 9 and Theorem 6 implies that 0 ∈ σp(T ). But this is not true, as T is injective. Thus 0 /∈ σ(T ) and T−1 is bounded totally paranormal operator, by Theorem 5. Hence T−1 is normaloid and [17, Proposition 2.12] implies that γ(T ) = 1 ∥T−1∥ = 1 r(T−1) = 1 sup {|µ| : µ ∈ σ(T−1)} = inf {|ν| : ν ∈ σ(T )} = d(0, σ(T )). This completes the proof. As a consequence of Theorem 9 we have the following result. Corollary 2. If T ∈ L(H) is a densely defined closed totally paranormal operator and ker(T ) = ker(T ∗), then γ(T ) = d(T ) := inf {|µ| : µ ∈ σ(T ) \ {0}}. Proof. Consider the operator S0 = T |ker(T )⊥ : D(T )∩ker(T )⊥ → ker(T )⊥. By Theorem 5 and Theorem 9, S0 is paranormal and γ(T ) = m(S0) = d(0, σ(S0)) = d(T ). This proves the result The following example shows the following facts: • Theorem 9 does not hold if ker(T ) ̸= ker(T ∗). • It is well known that the residual spectrum of a closed densely defined normal oper- ator is empty. But this is not true in the case of totally paranormal operators. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 12 of 20 Example 3. Let T : ℓ2(N) → ℓ2(N) be defined by T (x1, x2, · · · ) = (0, x1, 2x2, 3x3, · · · ), where D(T ) = { (x1, x2, · · · ) ∈ ℓ2(N) : ∑+∞ j=1 ∥jxj∥ 2 < +∞ } . As c0, the space of all complex sequences consisting of at most finitely many non zero terms is a subset of D(T ) and is dense in ℓ2(N), we can conclude that T is densely defined. Hence T ∗ is well defined. Note that T is a closed operator. We can show that T ∗(x1, x2, · · · ) = (x2, 2x3, 3x4, · · · ) with D(T ∗) = { (xn) ∈ ℓ2(N) : ∑+∞ j=2 ∥(j − 1)xj∥2 < +∞ } . For any x = (xn) ∈ D((T − λI)2) and λ ∈ C, we have ∥(T − λI)x∥2 = +∞∑ j=1 ∥(j − λ)xj∥2 ≤ +∞∑ j=1 (j + 1− λ)(j − λ) ∥xj∥2 ≤ +∞∑ j=1 ((j + 1− λ)(j − λ) ∥xj∥)2  1 2 +∞∑ j=1 ∥xj∥2  1 2 ≤ ∥∥(T − λI)2x ∥∥ ∥x∥ . Hence T is totally paranormal. Since ∥Tx∥ ≥ ∥x∥ for all x ∈ D(T ) and ∥Te1∥ = ∥e1∥, we get m(T ) = 1. Also it can be easily verified that T is injective, ran(T ) = ℓ2(N) \ span {e1} is closed but ran(T ) ̸= H, so 0 ∈ σ(T ). Hence d(0, σ(T )) = 0 ̸= 1 = m(T ). Now we will show that σ(T ) = C. To prove this, we show that T − µI is injective and ker(T − µi)∗ ̸= {0}, for all µ ∈ C. Let µ ∈ C \ {0} and (T − µI))x = 0 for some x = (xn) ∈ D(T ). Then (−µx1, x1 − µx2, 2x2 − µx3, · · · ) = 0. Equating component-wise we get x = 0. This implies that T −µI injective. Let y = (yn) ∈ D(T ∗) be such that (T − µI)∗y = 0. That is, (y2 − µ̄y1, 2y3 − µ̄y2, y4 − µ̄y3, · · · ) = 0. From this we get y = ( 1, µ̄, (µ̄)2 2! , (µ̄)3 3! , · · · ) y1. (3) If µ = 0, then ker(T ∗) = span{e1}. If µ ̸= 0, then we will show that y obtained in Equation (3) belongs to ker(T − µI)∗. Consider zn = µ̄2n (n!)2 . Then∣∣∣∣zn+1 zn ∣∣∣∣ = |µ|2 n+ 1 → 0 as n → +∞. S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 13 of 20 By the ratio test we conclude that ∑+∞ j=1 zn is absolutely convergent, that is ∑+∞ n=1 ( |µ|n n! )2 < +∞. Thus y ∈ ℓ2(N). On the similar lines we can show that ∑+∞ n=1 ( |µ|n (n−1)! )2 < +∞. Hence ker(T − µI))∗ ̸= {0}. For every µ ∈ C, ker(T − µI) = {0} and ran(T − µI) = (ker(T − µI)∗)⊥ ̸= ℓ2(N). Hence we conclude that µ ∈ σr(T ), and σ(T ) = C. We also have γ(T ) = 1 ̸= 0 = d(T ). From this we conclude that Corollary 2 is also not true if the condition, ker(T ) = ker(T ∗) is dropped. Theorem 10. Suppose T ∈ L(H) is a densely defined closed totally paranormal operator, ker(T ) = ker(T ∗) and 0 is an isolated spectral value of T . Then 0 ∈ σp(T ). Proof. Since 0 is an isolated spectral value of T , d(T ) > 0. Hence by Corollary 2, γ(T ) >) so that by Theorem 8, ran(T ) is closed. If 0 /∈ σp(T ), then ker(T ) = {0} so that we also have ran(T ) = ran(T ) = ker(T )⊥ = H, making T bijective and hence 0 /∈ σ(T ), a contradiction. Hence 0 ∈ σp(T ). Example 4. The converse of Theorem 10 need not be true. To see this, consider T : ℓ2(N) → ℓ2(N) defined by T (x1, x2, x3, · · · ) = ( 0, 2x2, 1 3 x3, 4x4, 1 5 x5, · · · ) , where D(T ) = { x ∈ ℓ2(N) : ( 0, 2x2, 1 3x3, 4x4, 1 5x5, · · · ) ∈ ℓ2(N) } . Here T is a densely de- fined closed totally paranormal operator. Since T is not one to one, 0 ∈ σp(T ) but it is not an isolated point of the spectrum σ(T ) = { 0, 2, 13 , 4, 1 5 , · · · } . Theorem 11. Suppose T ∈ L(H) is a densely defined closed totally paranormal operator, ker(T ) = ker(T ∗) . Then ran(T ) is closed if and only if 0 is not an accumulation point of σ(T ). Proof. By Theorem 2, ran(T ) is closed if and only if γ(T ) > 0 and by Corollary 2, γ(T ) = d(T ). Hence, ran(T ) is closed if and only if d(T ) > 0 if and only if 0 is not an accumulation point of σ(T ). 4. Weyl’s theorem for totally paranormal operators In this section, we demonstrate the fulfillment of Weyl’s theorem by a densely defined closed operator T that is totally paranormal. Additionally, we establish the self-adjointness of the Riesz projection Eµ corresponding to any non-zero isolated spectral value µ of T . For a Hilbert space H decomposed as H = H1⊕H2, where T ∈ L(H) is a closed operator, we ascertain the block matrix representation of T . T = [ T11 T12 T21 T22 ] (4) S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 14 of 20 where Tij : D(T ) ∩ Hj → Hi is defined by Tij = PHiTPHj |D(T )∩Hj for i, j = 1, 2. Here PHi is an orthogonal projection onto Hi. For (x1, x2) ∈ (H1 ∩D(T )⊕ (H2 ∩D(T )), T (x1, x2) = (T11x1 + T12x2, T21x1 + T22x2). Remark 3. Let T be as defined in Equation (4). If H1 = ker(T ) ̸= {0} and H2 = ker(T )⊥, then T = [ 0 T12 0 T22 ] . (5) • If T is densely defined closed operator then by Lemma 1, T22 ∈ L(ker(T )⊥) is also densely defined closed operator. • It can be easily checked that ran(T22) = ran(T ) ∩ ker(T )⊥. If ran(T ) is closed, then ran(T22) is closed in ker(T )⊥. We say a closed operator T ∈ L(H) satisfy the Weyl’s theorem if the Weyl’s spec- trum, σw(T ) consists of all spectral values of T except the isolated eigenvalues of finite multiplicity. That is, σ(T ) \ σw(T ) = π00(T ). Coburn [5] established that Weyl’s theorem applies to all bounded hyponormal and Toeplitz operators. Schmoeger [22] subsequently extended this finding to include bounded totally paranormal operators. In this study, we seek to verify the applicability of Weyl’s theorem to unbounded totally paranormal operators. Theorem 12. Let T ∈ L(H) be a densely defined closed totally paranormal operator. Then Weyl’s theorem holds for T , that is, σ(T ) \ σw(T ) = π00(T ). Proof. Let µ ∈ σ(T )\σw(T ). So, we have dim(ker(T−µI)) = dim(ker(T−µI)∗) < +∞ and ran(T − µI) is closed. On H = ker(T − µI)⊕ ker(T − µI)⊥, T − µI can be decomposed as T − µI = [ 0 T12 0 T22 − µI|ker(T−µI)⊥ ] , where T22 = P |ker(T−µI)⊥T |ker(T−µI)⊥ . By Remark 3, T22−µIker(T−µI)⊥ is a densely defined closed operator with domainD(T−µI)∩ker(T−µI)⊥ and ran(T22−µIker(T−µI)⊥) is closed. As ker(T −µI) is finite dimensional, this implies T12 is finite rank operator and by Remark 2, ind(T − µI) = ind ( T22 − µI|ker(T−µI)⊥ ) = 0. Since ker(T22 − µI|ker(T−µI)⊥) = {0} and ind ( T22 − µI|ker(T−µI)⊥ ) = 0, we get ker(T22 − µI|ker(T−µI)⊥) ∗ = {0} and consequently ran(T22 − µI|⊥ker(T−µI)) = ker(T − µI)⊥. Thus T22 − µI|⊥ker(T−µI) has bounded inverse and hence µ /∈ σ(T22). As σ(T ) ⊆ {µ} ∪ σ(T22), this implies that µ is an isolated point of σ(T ). Hence µ ∈ π00(T ). Conversely, let µ ∈ π00(T ). Now consider the Riesz projection Eµ with respect to µ. By Theorem 1 and S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 15 of 20 Theorem 7, µ /∈ σ(T |ker(Eµ)) and ran(T − µI) = ran ( (T − µI) |ker(Eµ) ) = ker(Eµ). Since µ /∈ σ(T |ker(Eµ)), we have that ran((T − µI)|ker(Eµ)) = ker(Eµ). Hence ran(T − µI) is closed. Also ((T − µI)|ker(Eµ)) −1 ∈ B (ker(Eµ)). Thus we get dimker(T − µI)∗ = dim ( ran(T − µI)⊥ ) = dim ( ker(Eµ) ⊥ ) = dim (ran(Eµ)) = dim (ker(T − µI)) . Note that dim ( ker(Eµ) ⊥) = dim (ran(Eµ)) but the spaces, ker(Eµ) ⊥ and ran(Eµ) need not be the same. Hence T −µI is Fredholm operator of index zero. This proves our result. Theorem 13. Let T ∈ L(H) be a densely defined closed totally paranormal operator and µ be a non-zero isolated point of σ(T ). Then the Riesz projection Eµ with respect to µ satisfy ran(Eµ) = ker(T − µI) = ker(T − µI)∗. Moreover, Eµ is self-adjoint. Proof. Let µ be a non-zero isolated point of σ(T ). By Theorem 1 and Theorem 7, µ /∈ σ ( T |ker(Eµ) ) and ran(T −µI) = ker(Eµ). That means (T −µI)|ker(Eµ) : ker(Eµ)∩D(T ) → ker(Eµ) = ran(T−µI) is a bijection. Also (T−µI)|ker(T−µI)⊥∩D(T ) : ker(T−µI)∩D(T ) → ran(T − µI) is a bijection, we have ker(Eµ) ∩D(T ) ⊆ ker(T − µI)⊥ ∩D(T ). Now we claim that ker(Eµ)∩D(T ) = ker(T −µI)⊥ ∩D(T ). Let x ∈ ker(T −µI)⊥ ∩D(T ) and Eµx = p+ q, where p ∈ ker(T − µI), q ∈ ker(T − µI)⊥. Operating Eµ on both sides, we get p+ q = Eµx = p+ Eµq. This implies Eµq = q ∈ ran(Eµ) ∩ ker(T − µI)⊥ = {0}, by Theorem 6. From this we conclude that Eµx = p = Eµp, that is, x−p ∈ ker(Eµ)∩D(T ) ⊆ ker(T −µI)⊥∩D(T ). As x ∈ ker(T − µI)⊥, we get p ∈ ker(T − µI) ∩ ker(T − µI)⊥ = {0}. Consequently Eµx = 0. So ker(T −µI)⊥∩D(T ) ⊆ ker(Eµ)∩D(T ). Hence ker(T −µI)⊥∩D(T ) = ker(Eµ)∩D(T ). By Lemma 1 and Theorem 7, we get ker(T − µI)⊥ = ker(T − µI)⊥ ∩D(T ) = ker(Eµ) ∩D(T ) = ran(T − µI) ∩D(T ) = (ker(T − µI)∗)⊥ ∩D(T ) ⊆ (ker(T − µI)∗)⊥. Hence ker(T − µI)∗ ⊆ ker(T − µI). By Theorem 7, ker(Eµ) ⊥ = ran(T − µI)⊥ = ker((T − µI)∗) ⊆ ker(T − µI) = ran(Eµ). Hence ker(Eµ) ⊥ = ran(T − µI)⊥ = ker((T − µI)∗) ⊆ ker(T − µI) = ran(Eµ). Hence ker(Eµ) ⊥ ⊆ ran(Eµ). If x ∈ ran(Eµ), then x = u + v where u ∈ ker(Eµ) and v ∈ ker(Eµ) ⊥. As ker(Eµ) ⊥ ⊆ ran(Eµ), we get u = x − v ∈ ker(Eµ) ∩ ran(Eµ) = {0}. Thus we get ker(Eµ) ⊥ = ran(Eµ), which is equivalent to say that ker(T−µI) = ker(T−µI)∗. As ker(Eµ) ⊥ = ran(Eµ), we have that Eµ is an orthogonal projection. Hence Eµ is self-adjoint. From the proof of Theorem 13, we have S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 16 of 20 Corollary 3. Let T ∈ L(H) be a densely defined closed paranormal operator. Then ker(T − µI)∗ ⊂ ker(T − µI) for all µ ∈ C. Applying the concept of Birkhoff–James orthogonality, we show that for a paranormal operator, the eigenspaces corresponding to distinct isolated eigenvalues are entirely inde- pendent of one another. Specifically, let M be a subspace within a Banach space X. We say that M is Birkhoff–James orthogonal to another subspace N in X if ∥m∥ ≤ ∥m+ n∥ for all m ∈ M and n ∈ N . This definition aligns with the concept of Birkhoff–James orthogonality, which, in the case of a Hilbert space X, matches the traditional notion of orthogonality. Proposition 6. Let T ∈ L(H) be a densely defined closed totally paranormal operator. If µ1 and µ2 are two non zero distinct isolated points of σ(T ), then ker(T−µ1I) is orthogonal to ker(T − µ2I). Proof. Without loss of generality, assume that |µ1| < |µ2|. For any x ∈ ker(T − µ1I) and y ∈ ker(T − µ2I), consider the set M = span{x, y}. As M is invariant subspace for T , it follows that T |M is totally paranormal operator and ∥T |M∥ = |µ2|. We have the following. ∥∥∥∥µn 1 µn 2 x+ y ∥∥∥∥ = 1 |µn 2 | ∥µn 1x+ µn 2y∥ ≤ ∥T |M∥n |µn 2 | ∥x+ y∥ = ∥x+ y∥ Taking the limit n → +∞, we get ∥y∥ ≤ ∥x+ y∥, for every x ∈ ker(T − µ1I) and y ∈ ker(T − µ2I). Hence ker(T − µ2I) is orthogonal to ker(T − µ1I). Next, if |µ1| = |µ2|, then for every n ∈ N∥∥∥∥(µ1 + µ2 2µ2 )n x+ y ∥∥∥∥ = ∥∥∥∥(µ1 + µ2) nx+ (µ1 + µ2) ny (2µ2)n ∥∥∥∥ ≤ 1 (2|µ2|)n n∑ j=0 ( n j ) |µ2|j ∥∥∥µn−j 1 x+ µn−j 2 y ∥∥∥ = 1 (2|µ2|)n n∑ j=0 ( n j ) |µ2|j ∥∥∥(T |M )n−j (x+ y) ∥∥∥ ≤ 1 (2|µ2|)n n∑ j=0 ( n j ) |µ2|n ∥x+ y∥ = ∥x+ y∥ . As µ1 ̸= µ2, we have ∣∣∣µ1+µ2 2µ2 ∣∣∣ < 1. Now as n → +∞ in the above inequality we get that ∥y∥ ≤ ∥x+ y∥. This proves the result S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 17 of 20 Proposition 7. Let T ∈ L(H) be a densely defined closed totally paranormal operator and T 2 be a compact operator. Then T is also compact and normal. Proof. Assume that T is a totally paranormal operator . Hence, ∥Tx∥2 ≤ ∥∥T 2x ∥∥ ∥x∥ for every x ∈ D(T 2). (6) Let {xm} ∈ H be weakly convergent sequence with limit 0 in D(T ). From the compactness of T 2 and the relation (6) we get the following relation: ∥Txm∥2 → 0, m → +∞. From the last relation it follows that T is compact. Since T is compact σ(T ) is finite set or countable infinite with 0 as the unique limit point of it. Let σ(T ) \ {0} = {λn} with |λ1| ≥ |λ2| ≥ · · · ≥ |λn| ≥ |λn+1| ≥ · · · ≥ 0, and λn → 0 (n → +∞). By the compactness of T or isoloidness of T , λn ∈ σp(T ) and dim ker(T − λn) < +∞ for all n. Since ker(T − λn) ⊂ ker(T − λn) ∗, M := ⊕+∞ n=1 ker(T − λn) reduces T , and T is of the form T = ( +∞⊕ n=1 λn ) ⊕ T ′ on H = M⊕M⊥. By the construction, T ′ is totally paranormal and σ(T ′) = {0} hence T ′ = 0. This shows that T = ( +∞⊕ n=1 λn ) ⊕ 0 and it is normal. Theorem 14. Let T ∈ L(H) be a densely defined closed totally paranormal operator with σw(T ) = {0}. Then T is a compact normal operator. Proof. By Theorem 12, T satisfy Weyl’s theorem and this implies that each element in σ(T ) \ σw(T ) = σ(T ) \ {0} is an eigenvalue of T with finite multiplicity, and is isolated in σ(T ). Hence σ(T ) \ {0} is a finite set or a countable set with 0 as its only accumulation point. Put σ(T ) \ {λn}, where λn ̸= λm whenever n ̸= m and {|λn|} is a non-increasing sequence. Since T is normaloid, we have |λ1| = ∥T∥ . By Corollary 3, we have (T−λ1I)x = 0 implies (T − λ1I) ∗x = 0. Hence ker(T − λ1I) is a reducing subspace of T . Let E1 be the orthogonal projection onto ker(T − λ1I). Then T = λ1I ⊕ T1 on H = ran(E1) ⊕ ran(I −E1). Since T1 is totally paranormal by Theorem 5 (i) and σp(T ) = σp(T1)∪ {λ1}, we have λ2 ∈ σp(T1). By the same argument as above, ker(T − λ2I) = ker(T1 − λ2I) is a finite dimensional reducing subspace of T which is included in ran(I − E1). Put E2 be the othogonal projection onto ker(T − λ2I). Then T = λ1E1 ⊕ λ2E2 ⊕ T2 on H = ran(E1)⊕ran(E2)⊕ran(I−E1−E2). By repeating above argument, each ker(T−λnI) S. Alnabulsi, M.H.M. Rashid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5587 18 of 20 is a reducing subspace of T and ∥∥∥∥∥T − n⊕ k=1 λkEk ∥∥∥∥∥ = ∥Tn∥ = |λn+1| → 0 as n → +∞. Here Ek is the orthogonal projection onto ker(T − λk) and T = ( n⊕ k=1 λkEk) ⊕ Tn on H = n⊕ k=1 ran(Ek)⊕ (1− n∑ k=1 ran(Ek). Hence T = +∞⊕ k=1 λkEk is compact and normal because each Ek is a finite rank orthogonal projection which satisfies EkEt = 0 whenever k ̸= t by Proposition 6 and λn → 0 as n → +∞. 5. Conclusion To conclude, this article provides a thorough analysis of the spectral properties of to- tally paranormal closed operators within Hilbert spaces. The study goes beyond standard constraints on boundedness, also considering closed symmetric operators. The initial focus was on establishing the non-emptiness of the spectrum for such opera- tors, accompanied by a characterization of closed-range operators based on the spectrum. Building on these foundational results, Weyl’s theorem was proven for densely defined closed totally paranormal operators. Specifically, it was demonstrated that the difference between the spectrum σ(T ) and the Weyl spectrum σw(T ) is precisely the set of isolated eigenvalues with finite multiplicities, denoted as π00(T ). The final section of the article explored the self-adjointness of the Riesz projection Eµ corresponding to any non-zero isolated spectral value µ of the operator T . The relation- ships ran(Eµ) = ker(T − µI) = ker(T − µI)∗ were established for this Riesz projection. Furthermore, it was shown that if a closed totally paranormal operator T has a Weyl spectrum σw(T ) = 0, then T qualifies as a compact normal operator. In terms of future work, potential avenues include exploring applications of these spec- tral properties in specific mathematical or physical contexts. Additionally, investigating the implications of these results on related areas of operator theory or functional analy- sis could provide valuable insights. 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