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An operator $T$ acting on a Banach space $\X$ obeys property $(R)$ if $\pi_a^0(T)=E^0(T),$ where $\pi_a^0(T)$ is the set of all left poles of $T$ of finite rank and $E^0(T)$ is the set of all isolated eigenvalues of $T$ of finite multiplicity. In this paper we introduce and study two new properties $(S)$ and $(gS)$ in connection with Weyl type theorems. Among other things, we prove that if $T$ is a bounded linear operator acting on a Banach space, then $T$ satisfies property $(R)$ if and only if $T$ satisfies property $(S)$ and $\pi^0(T)=\pi_a^0(T),$ where $\pi^0(T)$ is the set of poles of finite rank. Also we show that Weyl's theorem holds for $T$ if and only if $T$ satisfies property $(S)$ and $\sigma_w(T)=\sigma_b(T)$, where $\sigma_w(T)$ is the Weyl spectrum and $\sigma_b(T)$ is the the Browder spectrum. Analogous results for property $(gS)$ are given. Moreover, these properties are also studied in the frame of polaroid operator.

In this paper, wefind necessary and sufficient conditions for Banach Space operator to satisfy the property(Bb). Thenweobtain,ifBanachSpaceoperatorsA ∈ B(X)andB ∈ B(Y)satisfyproperty(Bb)implies A⊗Bsatisfiesproperty(Bb)ifandonlyiftheB-Weylspectrumidentity σBW(A⊗B) = σBW(A)σ(B)∪σBW(B)σ(A) holds. Perturbations by Riesz operators are considered. Throughout this paper we denote by B(X) the algebra of all bounded linear operators acting on a Banach space X. For T ∈ B(X), let T ∗ ,ker(T) = T −1 (0), ℜ(T) = T(X), σ(T) and σa(T) denote respectively the adjoint, the null space, the range, the spectrum and the approximate point spectrum of T. Let α(T) and β(T) be the nullity and the deficiency of T defined by α(T) = dimker(T) and β(T) = codimℜ(T). If the range ℜ(T) of T ∈ B(X) is closed and α(T) < ∞ (resp., β(T) < ∞) then T is upper semi-Fredholm (resp., lower semi-Fredholm) operator. Let SF+(X) (resp.,SF−(X)) denote the semigroup of upper semi- Fredholm (resp., lower semi-Fredholm) operator on X. An operator T ∈ B(X) is said to be semi-Fredholm if T ∈ SF+(X) ∪ SF−(X) and Fredholm if T ∈ SF+(X) ∩ SF−(X). If T is semi-Fredholm then the index of T is defined by ind(T) = α(T) − β(T). Recall that the ascent of an operator T ∈ B(X) is the smallest non negative integer p:=p(T) such that T −p (0) = T −(p+1) (0). If there is no such integer, ie., T −p (0) , T −(p+1) (0) for all p, then set p(T) = ∞ . The descent of T is defined as the smallest non negative integer q:=q(T) such that T q (X) = T (q+1) (X). If there is no such integer, ie., T q (X) , T (q+1) (X) for all q, then set q(T) = ∞. It is well known that if p(T) and q(T) are both finite then they are equal (13, Proposition 38.6). A bounded linear operator T acting on a Banach space X is Weyl if it is Fredholm of index zero and Browder if T is Fredholm of finite ascent and descent. For T ∈ B(X), let , E 0 (T), and π 0 (T) denote, the eigenvalues of finite multiplicity and poles of T respectively . The Weyl spectrum σw(T) and Browder spectrum σb(T) of T are defined by σw(T) = { λ ∈ C : T − λ is not Weyl } , σb(T) = {λ ∈ C : T − λ is not Browder } . We have π 0 (T) := σ(T) \ σb(T). Set ∆(T) = σ(T) \ σw(T). According to Coburn (7), Weyl's theorem holds for T (abbreviation, T ∈ Wt) if ∆(T) = E 0 (T) and that Browder's theorem holds for T (in symbol, T ∈ Bt) if σ(T) \ σw(T) = π 0 (T).