Putnam Theorem Research Articles

Generalizations of Weiss’ version of the Fuglede–Putnam theorem for unbounded operators

The present paper discusses Weiss’ version of the Fuglede–Putnam theorem. We establish three generalizations of this version, each of which involves unbounded normal operators. The first generalization expands upon Weiss’ theorem by incorporating a single unbounded normal operator, while the second involves normal powers. The third generalization deals with two unbounded normal operators. These generalizations have several applications and are accompanied by counterexamples.

Asymmetric Fuglede–Putnam theorem for unbounded M-hyponormal operators

A closed densely defined operator T on a Hilbert space \({\mathcal {H}}\) is called M-hyponormal if \({\mathcal {D}}(T) \subset {\mathcal {D}}(T^{*})\) and there exists \(M>0\) for which \(\Vert (T-zI)^{*}x \Vert \le M \Vert (T-zI)x \Vert \) for all \(z \in {\mathbb {C}}\) and \(x\in {\mathcal {D}}(T)\). In this paper, we prove that if \(A:{\mathcal {H}}\rightarrow {\mathcal {K}}\) is a bounded linear operator, such that \(AB^*\subseteq TA\), where B is a closed subnormal (resp. a closed M-hyponormal) on \({\mathcal {H}}\), T is a closed M-hyponormal (resp. a closed subnormal) on a Hilbert space \({\mathcal {K}}\), then (i) \( AB\subseteq T^*A\) (ii) \({\overline{\text{ ran }\,A^{*}}}\) reduces B to the normal operator \( B\vert _{{\overline{\text{ ran }\,A^{*}}}}\) and (iii) \({\overline{\text{ ran }\,A}}\) reduces T to the normal operator \( T\vert _{{\overline{\text{ ran }\,A}}}\).

Basic properties of certain class of non normal operators

Some properties of k-quasi-M-hyponormal are established in this paper. The ascent and an extension of thewell-known Fuglede Putnam's Theorem for such operators as well as other related results are also presented,which complete some results given in [7, 12].

A generalized Fuglede–Putnam theorem for a class of almost normal tuples of operators with finite modulus of Hilbert–Schmidt quasitriangularity

We extend the Fuglede–Putnam theorem modulo the Hilbert– Schmidt class to almost normal m-tuples of operators with finite Hilbert– Schmidt modulus of quasitriangularity.

The elementary operator and Fuglede–Putnam theorem for a class of essentially commuting almost normal operators with finite modulus of Hilbert–Schmidt quasitriangularity

The elementary operator and Fuglede–Putnam theorem for a class of essentially commuting almost normal operators with finite modulus of Hilbert–Schmidt quasitriangularity

Fuglede–Putnam theorem for $$(\alpha , \beta )$$-normal operators

In this paper, we obtain some properties of $$(\alpha , \beta )$$ -normal and we prove following assertions.

Fuglede\u2013Putnam type theorems for (p,k)-quasihyponormal operators via hyponormal operators

For Hilbert space operators S, X, and T, (S,X,T)in FP means Fuglede–Putnam theorem holds for triplet (S,X,T), that is, SX=XT ensures S^{ast }X=XT^{ast }. Similarly, (S,T)in FP means (S,X,T)in FP holds for each operator X. This paper is devoted to the study of Fuglede–Putnam type theorems for (p,k)-quasihyponormal operators via a class of operators based on hyponormal operators FP(H):={S|(S,T)in FP mbox{ holds for each hyponormal operator } T^{ast }}. Fuglede–Putnam type theorems involving (p,k)-quasihyponormal, dominant, and w-hyponormal operators, which are extensions of the results by Tanahashi, Patel, Uchiyama, et al., are obtained.

On commutator of generalized Aluthge transformations and Fuglede–Putnam theorem

Let $$A=U|A|$$ be the polar decomposition of A on a complex Hilbert space $${\mathscr {H}}$$ and $$0<s,t$$ . Then $${\widetilde{A}}_{s, t}=|A|^sU|A|^t$$ and $${\widetilde{A}}_{s, t}^{(*)}=|A^*|^sU|A^*|^t$$ are called the generalized Aluthge transformation and generalized $$*$$ -Aluthge transformation of A, respectively. A pair (A, B) of operators is said to have the Fuglede–Putnam property (breifly, the FP-property) if $$AX=XB$$ implies $$A^*X=XB^*$$ for every operator X. We prove that if (A, B) has the FP-property, then $$({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})$$ and $$(({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})$$ has the FP-property for every $$s,t>0$$ with $$s+t=1$$ . Also, we prove that $$({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})$$ has the FP-property if and only if $$(({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})$$ has the FP-property, where A, B are invertible and $$ 0 < s, t $$ with $$ s + t =1$$ . Moreover, we prove that if $$0 < s, t$$ and $${\widetilde{A}}_{s, t}$$ is positive and invertible, then $$\left\| {\widetilde{A}}_{s, t}X-X{\widetilde{A}}_{s, t}\right\| \le \left\| A\right\| ^{2t}\left\| ({\widetilde{A}}_{s, t})^{-1}\right\| \left\| X\right\| $$ for every operator X. Also, if $$ 0 <s, t$$ and X is positive, then $$\left\| |{\widetilde{A}}_{s, t}|^{2r} X-X|{\widetilde{A}}_{s, t}|^{2r}\right\| \le \frac{1}{2}\left\| |A|\right\| ^{2r}\left\| X\right\| $$ for every $$r>0$$ .

Fuglede–Putnam theorem in algebras with involutions

The validity of the analogs of the Fuglede–Putnam theorem in the algebra (B(H), ٭) of bounded operators acting on a Hilbert space H with an arbitrary involution ٭ is considered, together with the same problem in certain *-subalgebras of these algebras and in related constructions. The results obtained in this way are used to solve stability problems for “Fuglede” classes with respect to extensions and to the operation of tensor products.

Fuglede–Putnam theorem for class A operators

Fuglede–Putnam theorem for class A operators

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that \(D(N)\subseteq D(A)\). Then, \( AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.\)

An All-Unbounded-Operator Version of the Fuglede–Putnam Theorem

We present in this paper a new version of the famous Fuglede–Putnam theorem where all the operators involved are unbounded.

YET MORE VERSIONS OF THE FUGLEDE–PUTNAM THEOREM

AbstractWe give two types of generalisation of the well-known Fuglede–Putnam theorem. The paper is ‘spiced up’ with some examples and applications.

Another Version of Fuglede–Putnam Theorem

Abstract In [Yoshino, Proc. Amer. Math. Soc. 95: 571–572, 1985] the author proved that for a 𝑀-hyponormal operator 𝐴* and for a dominant operator 𝐵, 𝐶𝐴 = 𝐵𝐶 implies 𝐶𝐴* = 𝐵*𝐶. In the case where 𝐴* and 𝐵 are normal, this result is known as the Fuglede–Putnam theorem. In this paper, we will extend this result to the case in which 𝐴 is an injective (𝑝, 𝑘)-quasihyponormal operator and 𝐵* is a dominant operator. We also show that the same result remains valid for (𝑝, 𝑘)-quasihyponormal and log-hyponormal operators.

On the converse of Anderson’s theorem

We investigate the converse of Anderson’s theorem on the range-kernel orthogonality of a derivation. In particular, we show that a pair of bounded linear operators on a Hilbert space satisfies the Fuglede–Putnam theorem relative to the ideal of compact operators if and only if it satisfies Anderson’s inequality relative to the same ideal.

Real realization: Dennett’s real patterns versus Putnam’s ubiquitous automata

Both Putnam and Searle have argued that that every abstract automaton is realized by every physical system, a claim that leads to a reductio argument against Cognitivism or Strong AI: if it is possible for a computer to be conscious by virtue of realizing some abstract automaton, then by Putnam's theorem every physical system also realizes that automaton, and so every physical system is conscious--a conclusion few supporters of Strong AI would be willing to accept. Dennett has suggested a criterion of reverse engineering for identifying "real patterns," and I argue that this approach is also very effective at identifying "real realizations." I focus on examples of real-world implementations of complex automata because previous attempts at answering Putnam's challenge have been overly restrictive, ruling out some realizations that are in fact paradigmatic examples of practical automaton realization. I also argue that some previous approaches have at the same time been overly lenient in accepting counter-intuitive realizations of trivial automata. I argue that the reverse engineering approach avoids both of these flaws. Moreover, Dennett's approach allows us to recognize that some realizations are better than others, and the line between real realizations and non-realizations is not sharp.

Generalized Derivation and Double Operator Integrals

Abstract Let 𝐻 be a separable infinite dimensional complex Hilbert space, and let 𝔹(𝐻) denote the algebra of all bounded linear operators on 𝐻. Let 𝐴, 𝐵 be operators in 𝔹(𝐻). We define the generalized derivation δ 𝐴, 𝐵 : 𝔹(𝐻) ↦ 𝔹(𝐻) by δ 𝐴, 𝐵(𝑋) = 𝐴𝑋 – 𝑋𝐵. In this paper we consider the question posed by Turnsek [Publ. Math. Debrecen 63: 293–304, 2003], when ? We prove that this holds in the case where 𝐴 and 𝐵 satisfy the Fuglede–Putnam theorem. Finally, we apply the obtained results to double operator integrals.

On q–deformed hyponormal operators

AbstractThe present paper deals with operators in a Hilbert space related to the theory of quantum groups. Some results on q–deformed hyponormal operators extend ones on q–normal operators. For a q–deformed operator T the Cartesian decomposition of the inverse T–1 is characterized and the product BT with bounded operator B is analyzed, and a q–analogue of the Fuglede–Putnam theorem is discussed.

A class of hyponormal operators and weak*-continuity of hermitian operators

We will first in this paper consider a class of hyponormal operators which we call *-hyponormal operators. We give an example of a hyponormal operator which is not *-hyponormal. It follows from a theorem of Ackermans, van Eijndhoven and Martens [1] that subnormal operators on a Hilbert space are *-hyponormal. We prove a generalized Fuglede--Putnam theorem and some other results for these operators. We will also prove some results on the following problem which was mentioned in [4]:

Putnam's theorem, Alexander's spectral area estimate, and VMO

In this paper we show that ] f f is a bounded analytic function defined on the unit disk such that at each point of the unit circle the cluster set o f f has area zero, then f has vanishing mean oscillation (see Sect. 1 for definitions). We discovered this result (Corollary 1.5) and its quantitative version (Theorem 1.4) using some techniques from operator theory. This proof is given in Sect. 1; the main tool is Putnam's Theorem (Lemma 1.1). In the main result in Sect. 2 (Theorem 2.4) we extend the above result to the unit ball in ~ ' , using commutative Banach algebra techniques rather than operator theory. Here the key tool is Alexander's spectral area estimate (Lemma 2.2). Section 3 ties the techniques of Sects. 1 and 2 together by giving a simple proof 0fPutnam's Theorem for subnormal operators. Here the main tool (Lemma 3.2) is a quantitative version of the Hartogs-Rosenthal Theorem. Some concluding remarks indicate another method of combining the tools used in this paper. Although there are obvious connections between the three sections of the paper, each section can be read independently. Throughout the paper, c denotes an absolute constant, independent of everything except the dimension n of ~n (in Sect. 2). However, the actual value of c may change; thus sometimes c 2 is replaced by c.