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There are several nonequivalent extensions of orthogonality from inner product spaces to general normed spaces. One of the most well-known is the Birkhoff-James orthogonality: if X is a normed space and x, y∈X then x is Birkhoff-James orthogonal to y if ‖x+λy‖≥‖x‖ for all scalars λ. In a C*-algebra A we can also discuss the case when x∈A is Birkhoff-James orthogonal to all the elements of the form ya, a∈A. In this case we say that x is strongly Birkhoff-James orthogonal to y. We discuss this kind of orthogonality in some special classes of C*-algebras. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.

U ovom članku navodimo nekoliko zanimljivih zadataka iz determinanti, te geometrijski interpretiramo elementarne transformacije nad determinantama u slučaju determinanti drugog reda. Nadalje, dajemo geometrijsku interpretaciju tvrdnje da je determinanta matrice reda 2 jednaka determinanti njene transponirane matrice.

General normed spaces are not equipped with an inner product. Nonetheless, there do exist several nonequivalent extensions of orthogonality from inner product spaces to general normed ones. One of the most well-known is the Birkhoff-James orthogonality: if X is a normed space and x, y elements in X, then x is Birkhoff--James orthogonal to y if the distance from x to onedimensional space spanned by y is equal to the norm of x. In a C*-algebra A we can also discuss the case when x is Birkhoff--James orthogonal to all the elements of the form ya, a\in A. In this case we say that x is strongly Birkhoff-James orthogonal to y. We discuss this kind of orthogonality in commutative C*-algebras. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.

We classify surjective linear maps on B(H) that preserve mutual strong Birkhoff--James orthogonality.

In this talk we present some results on mutual strong Birkhoff–James orthogonality in two classical C*-algebras: the C*-algebra B(H) of all bounded linear operators on a complex Hilbert space H and a commutative, possibly nonunital, C*-algebra. With the help of the induced graph it is shown that this relation alone can characterize right invertible elements. Moreover, in the case of commutative unital C*-algebras, it can detect the existence of a point with a countable local basis in the corresponding compact Hausdorff space. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.

We say that two elements of a Hilbert C*-module are orthogonal if their C*-valued inner product is 0. In a Hilbert C*-module, besides this type of orthogonality, we can study all other orthogonalities defined in a general normed space. One which is most frequently used is Birkhoff–James orthogonality - if x, y are elements of a normed linear space X, then x is orthogonal to y in the BJ sense if ∥x + λy∥ ≥ ∥x∥ for all scalars λ. As we usually do in Hilbert C*-modules, we study analogous relations obtained by replacing scalars with elements of the underlying C*- algebra, or the norm with the C*- valued ”norm”. It often happens that these relations are very strong and coincide with (the first mentioned) orthogonality in a Hilbert C*- module, but not always. This leads to the notion of the strong (also called modular) BJ orthogonality which is the main topic of this talk. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.

Za proizvoljnu matricu reda 2 promatramo prostor generiran njenim potencijama, te otkrivamo njegovu dimenziju. Rješavanjem ovog zadatka dolazimo do karakterističnog polinoma i Hamilton-Cayleyjevog teorema za matrice reda 2.

We give a complete description of 2x2 complex matrices A and B which are Roberts orthogonal to each other. Necessary and sufficient conditions for Roberts orthogonality of A and B are also given in terms of traces of matrices.

Let (x_n) be a frame for a Hilbert space H. We say that a finite set of indices E satisfies the minimal redundancy condition for (x_n) if the subset {; ; x_n : n ∈ N\E}; ; is dense in H. Assuming E has the minimal redundancy property we present a method for construction of a dual frame for a reduced sequence starting from a dual frame for (x_n). The talk is based on joint works with Damir Bakić and Diana Stoeva.

A frame for an N-dimensional Hilbert space H is said to be a full spark frame if any N of its vectors form a basis for H. In this talk we present a characterization of full spark frames in terms of certain totally nonsingular matrices. Also, we construct a family of infinite totally positive matrices which are a subclass of the class of totally nonsingular matrices. This is a joint work with Damir Bakić.

U ovom radu diskutiramo o nizovima vektora u dvodimenzionalnom prostoru koji omogućuju jednostavnu rekonstrukciju svih vektora prostora.

U ovom članku riješit ćemo dva zadataka koja se bave problemom najkraćeg puta između dviju točaka ravnine.

We say that the elements x and y of a complex normed space X are Roberts orthogonal if the norm of x + λy is the same as the norm of x − λy for all complex numbers λ. Let A be a unital C*- algebra with the unit e. In this talk we present results which characterize Roberts orthogonality of the unit e and an arbitrary element a of A. This is a joint work with Tomislav Berić and Rajna Rajić.

Let H be an N-dimensional Hilbert space. A frame for H is said to be a full spark frame if any N of its members make up a basis for H. In this talk we present a connection between full spark frames and totally nonsingular matrices. We also provide a method for constructing infinite totally positive matrices which are a subclass of the class of totally nonsingular matrices. This is a joint work with Damir Bakić.

Two vectors in an inner product space are orthogonal if their inner product is 0. There are many different ways how the notion of orthogonality can be generalized to the class of normed spaces. Among them, the Birkhoff--James orthogonality and the Roberts orthogonality are the most important. In this talk we shall present some basic results about these two types of orthogonality. Also, we shall present some characterizations in $C^*$- algebras which are normed spaces with some additional structures. This is a joint work with Rajna Rajić and Tomislav Berić.

A sequence in a Hilbert space H is a frame for H if there exist positive constants A and B such that, for every unit x in H, the sum of squares of inner products of x with elements of frames converges to the number between A and B. Frames are often used in process of encoding and decoding signals. It is the redundancy property of frames that makes them useful in applications where some of the coefficients could be lost during the data transmission. We discuss the problem of recovering signal from frame coefficients with erasures. Provided that the erasure set satisfies the minimal redundancy condition, we construct a suitable synthesizing dual frame which enables a perfect reconstruction of the original signal without recovering the lost coefficients. This is a joint work with Damir Bakić.

Let (x_n) be a frame for a Hilbert a-module X. Its synthesis operator is a surjective adjointable map from \ell^2(a) to X. If a is a unital C*-algebra, then there is a bijective correspondence of the set of all adjointable surjections from \ell^2(A) to X and the set of all frames for X. In the non-unital case this does not hold ; namely, some adjointable surjections from \ell^2(A) to X are not synthesis operators of any frame for X. However, it turns out that such operators produce frame-like sequences that we call outer frames for X. This is a joint work with Damir Bakić.

Let B(H) be the algebra of all bounded linear operators acting on a complex Hilbert space H. Linear preserver problems on B(H) concern the characterizations of all linear mappings on B(H) that leave certain functions, subsets, relations, etc. invariant We characterize the class of surjective (conjugate) linear mappings on B(H) that preserve the (strong) Birkhoff-James orthogonality in both directions. We also describe mappings on B(H) preserving the orthogonality of ranges of Hilbert space operators in terms of the Roberts orthogonality.

Orthogonality in normed linear spaces can be defined in different ways. We consider two types of orthogonality in C*-algebras. The Birkhoff-James orthogonality for arbitrary elements a and b of a C*-algebra A is characterized in terms of states acting on A. A characterization of a special case of the Roberts orthogonality for elements a and e, where e is the unit in A, is obtained in terms of the Davis-Wielandt shell of a.

A Hilbert C*-module over a C*-algebra A is a right A-module X equipped with an A-valued inner product such that X is a Banach space with respect to the norm induced from A. We discuss frames in X. In this talk we present some new results in the theory of frames for countably generated Hilbert C*-modules over arbitrary C*- algebras. In investigating the nonunital case we introduce the concept of an outer frame for X - in comparison with frames for X the difference is that the elements of an outer frame for X are merely members of a multiplier module M(X) and need not belong to X. Outer frames behave very much like frames ; for example, the reconstruction property for elements of X is preserved, and the only adjointable operators on M(X) that preserve the class of frames and outer frames for X are the surjective ones. Building on a unified approach to frames and outer frames we obtain new results on dual frames, frame perturbations, tight approximations of frames and finite extensions of Bessel sequences. This is a joint work with Damir Bakić.

A Hilbert C*-module over a C*-algebra A is a right A-module X equipped with an A-valued inner product such that X is a Banach space with respect to the norm induced from the norm on A. Hilbert C*-modules generalize Hilbert spaces, as well as C*-algebras. In the modular frame theory one of the most important Hilbert C*-modules is the generalized Hilbert space over A. Modular frames were introduced by M. Frank and D. Larson. In their papers they showed that many of the most important results from the Hilbert space frame theory still hold. The concept of outer frames for Hilbert C*-modules was introduced by Arambašić and Bakić - the only difference between frames and outer frames is that some of the vectors of outer frames belong to the multiplier module of X and not to X. In this talk we present some results on outer frames. This is a joint work with Damir Bakić.